Research Article The Entropy of Weighted Graphs with Atomic Bond Connectivity Edge Weights Young Chel Kwun , 1 Hafiz Mutee ur Rehman, 2,3 Muhammad Yousaf, 4 Waqas Nazeer , 2 and Shin Min Kang 5,6 1 Department of Mathematics, Dong-A University, Busan 49315, Republic of Korea 2 Division of Science and Technology, University of Education Lahore 54000, Pakistan 3 Department of Mathematics and Statistics, e University of Lahore, Lahore 54000, Pakistan 4 Department of Mathematics, COMSATS University Islamabad, Lahore, Campus, Lahore 54000, Pakistan 5 Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Republic of Korea 6 Center for General Education, China Medical University, Taichung 40402, Taiwan Correspondence should be addressed to Waqas Nazeer; [email protected] and Shin Min Kang; [email protected] Received 24 July 2018; Revised 30 October 2018; Accepted 27 November 2018; Published 16 December 2018 Academic Editor: Manuel De la Sen Copyright © 2018 Young Chel Kwun et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e aim of this report to solve the open problem suggested by Chen et al. We study the graph entropy with ABC edge weights and present bounds of it for connected graphs, regular graphs, complete bipartite graphs, chemical graphs, tree, unicyclic graphs, and star graphs. Moreover, we compute the graph entropy for some families of dendrimers. 1. Introduction Mathematical chemistry is the branch of theoretical chemistry in which we discuss and predict the behavior of mathemat- ical structure by using mathematical tools [1, 2]. In the past few decades, there have been many studies in this area. is theory has played an important role in the field of chemistry. e topological index is a real number associated with the molecular graph. It is a graph invariant. Many topological indices are defined up till now [3–5]. Some of them are based on distance, while others are based on degree and have found many applications in pharmacy, theoretical chemistry, and especially QSPR/QSAR research. In 1975, the first degree-based topological index [6] was proposed. Later, this index was generalized to any real number by Estrada et al. in [7] and was named the generalized Randi´ c index. Another well-known topological index based on the vertex degree of the graph is the atomic bond connectivity index [8], which is defined as () = ∑ V∈() √ + −2 . (1) At the beginning, a close link between the heat of alkane formation and the ABC index was experienced. Aſter that, the ABC index became a powerful tool for simulating the thermodynamic properties of organic compounds. e fourth member of the ABC index category was proposed by M. Ghorbani et al. in [9]. In recent years, many papers are written on topological indices and its application; here we mention few [10–15]. Based on the groundbreaking work of Shannon [16], in the late 1950s began to study the entropy measurement of network systems. Rashevsky uses the concept of graph entropy to measure the structural complexity of the graph. Here, the complexity of his graph is based on Shannon’s entropy. Mowshowitz [17] introduced the entropy of the graph as information theory, which he interpreted as the structural information content of the graph. Mowshowitz [18] later studied the mathematical properties of graph entropy and conducted indepth measurements of his particular application. Graph entropy measures have been used in various disciplines, for example, to characterize patterns in Hindawi Discrete Dynamics in Nature and Society Volume 2018, Article ID 8407032, 10 pages https://doi.org/10.1155/2018/8407032
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Research ArticleThe Entropy of Weighted Graphs with Atomic BondConnectivity Edge Weights
Young Chel Kwun 1 Hafiz Mutee ur Rehman23 Muhammad Yousaf4
Waqas Nazeer 2 and Shin Min Kang 56
1Department of Mathematics Dong-A University Busan 49315 Republic of Korea2Division of Science and Technology University of Education Lahore 54000 Pakistan3Department of Mathematics and Statistics The University of Lahore Lahore 54000 Pakistan4Department of Mathematics COMSATS University Islamabad Lahore Campus Lahore 54000 Pakistan5Department of Mathematics and RINS Gyeongsang National University Jinju 52828 Republic of Korea6Center for General Education China Medical University Taichung 40402 Taiwan
Correspondence should be addressed to Waqas Nazeer nazeerwaqasueedupk and Shin Min Kang smkanggnuackr
Received 24 July 2018 Revised 30 October 2018 Accepted 27 November 2018 Published 16 December 2018
Academic Editor Manuel De la Sen
Copyright copy 2018 Young Chel Kwun et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The aim of this report to solve the open problem suggested by Chen et alWe study the graph entropy with ABC edge weights andpresent bounds of it for connected graphs regular graphs complete bipartite graphs chemical graphs tree unicyclic graphs andstar graphs Moreover we compute the graph entropy for some families of dendrimers
1 Introduction
Mathematical chemistry is the branch of theoretical chemistryin which we discuss and predict the behavior of mathemat-ical structure by using mathematical tools [1 2] In the pastfew decades there have been many studies in this area Thistheory has played an important role in the field of chemistry
The topological index is a real number associated withthemolecular graph It is a graph invariant Many topologicalindices are defined up till now [3ndash5] Some of them are basedon distance while others are based on degree and have foundmany applications in pharmacy theoretical chemistry andespecially QSPRQSAR research
In 1975 the first degree-based topological index [6]was proposed Later this index was generalized to any realnumber 120572 by Estrada et al in [7] and was named thegeneralized Randic index Another well-known topologicalindex based on the vertex degree of the graph is the atomicbond connectivity index [8] which is defined as
119860119861119862 (119866) = sum119906Visin119864(119866)
radic119889119906 + 119889119906 minus 2119889119906119889119906 (1)
At the beginning a close link between the heat of alkaneformation and the ABC index was experienced After thatthe ABC index became a powerful tool for simulatingthe thermodynamic properties of organic compounds Thefourth member of the ABC index category was proposed byM Ghorbani et al in [9] In recent years many papers arewritten on topological indices and its application here wemention few [10ndash15]
Based on the groundbreaking work of Shannon [16]in the late 1950s began to study the entropy measurementof network systems Rashevsky uses the concept of graphentropy to measure the structural complexity of the graphHere the complexity of his graph is based on Shannonrsquosentropy Mowshowitz [17] introduced the entropy of thegraph as information theory which he interpreted as thestructural information content of the graphMowshowitz [18]later studied the mathematical properties of graph entropyand conducted indepth measurements of his particularapplication Graph entropy measures have been used invarious disciplines for example to characterize patterns in
HindawiDiscrete Dynamics in Nature and SocietyVolume 2018 Article ID 8407032 10 pageshttpsdoiorg10115520188407032
2 Discrete Dynamics in Nature and Society
biology chemistry and computer science Therefore it is notsurprising to realize that ldquograph entropyrdquo has been defined invarious ways Another classic example is the introduction ofKorners entropy [19]
Different graph invariances have been used to developimage entropy measures such as eigenvalue and connectivityinformation [20] distance-based graph entropy [21] anddegree-based graph entropy [22]
We have different applications of graph entropy in com-munications and economics We use the concept of graphentropy as a weighted graph just as Dehmer [20] who solvedthe problem of weighted chemical graph entropy by usinga special information functional Some degree-based indicesare characterized by investigating the extremes of the entropyof certain class of graphs [23 24]
In [25] Chen et al introduced the concept of graphentropy for special weighted graphs by using Randic edgeweights and proved extremal properties of graph entropyfor some elementary families of graphs Our aim is tosolve problem suggested by Chen et al in [25] In thispaper we study graph entropy by taking Atomic bondconnectivity edge weights and prove some extermal proper-ties of graph entropy for special families of graphs such asconnected graphs regular graphs complete bipartite graphschemical graphs tree unicyclic graphs and star graphsMoreover we compute graph entropy of different dendrimerstructures
2 Main Results
Let us have a graph where G and V119894 are its vertices and thedegree of V119894 is denoted by 119889119894 For an edge V119894V119895 we have
119901119894119895 = 119908 (V119894V119895)sum119889119894119895=1 119908(V119894V119895) (2)
where119908(V119894V119895) is the weight V119894V119895 and119908(V119894V119895)gt 0Theweightedentropy is defined as
119867(V119894) = minus 119889119894sum119895=1
119901119894119895 log (119901119894119895) (3)
Inspired by this method we have introduced the definitionof the entropy of the edge-weighted graph which can alsobe interpreted as multiple graphs For edge weight graphsG = (V E w) where V E and w denote the vertex sets ofG edge sets and edge weights (sometimes also called costs)In this article we always assume that the edge weights arepositive
Definition 1 Let 119866 = (119881 119864 119908) be an edge weighted then theentropy of G is
119868 (119866 119908) = minus sum119906Visin119864
119901119906V log (119901119906V) (4)
where 119901119906V = 119908(119906V)sum119906Visin119864119908(119906V)Theorem 2 For a connected graph G with 119899 vertices for 119899 ge 3we have
log (119860119861119862) + logradic 1119899 minus 2 le 119868 (119866 119860119861119862)le log (119860119861119862) minus log 1119899 minus 1
(5)
Proof For a simple connected graphof order 119899 themaximumdegree for a vertex is n-1 and minimum degree is 1 With anyedge uv the minimum possible degrees of u and v are 1 and2 respectively and maximum possible degrees of u and v aren-1 and n-1 so we have
119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sumradic119889119906 + 119889119906 minus 2119889119906119889119906sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (119860119861119862)minus 12 (119860119861119862) sumradic119889119906 + 119889119906 minus 2119889119906119889119906sdot [log(radic119889119906 + 119889V minus 2) minus log (119889119906119889V) )]le log (119860119861119862) minus 12 [log (1) minus log (119899 minus 1)2]= log (119860119861119862) minus logradic 1(119899 minus 1)2 le log (119860119861119862)minus log 1119899 minus 1
I (GABC) ge log (119860119861119862) minus 12 (119860119861119862) sumradic119889119906 + 119889119906 minus 2119889119906119889119906sdot [log (2119899 minus 4) minus log (2)] = log (119860119861119862)minus 12 log (119899 minus 2) = log (119860119861119862) + logradic 1119899 minus 2= log (119860119861119862) + logradic 1119899 minus 2
(6)
Corollary 3 Let G be a graph with 119899 vertices Let 120575 119886119899119889 Δ bethe minimum degree and maximum degree of G respectivelyThen we have
Discrete Dynamics in Nature and Society 3
log (119860119861119862) + log(radic 1205752Δ minus 2) le I (GABC)le log (119860119861119862) minus 12 log( Δradic2120575 minus 2)
(7)
Proof
I (GABC) le log (119860119861119862)minus 12 (119860119861119862) sumradic119889119906 + 119889V minus 2119889119906119889V [log (2120575 minus 2) minus log (Δ2)]= log (119860119861119862) minus 12 log( Δradic2120575 minus 2)
(8)
Also
I (GABC) ge log (119860119861119862) minus 12 [log (2Δ minus 2) minus log (1205752)]= log (119860119861119862) + log(radic 1205752Δ minus 2) (9)
Theorem 4 For a regular graph 119866 = (119881 119864 119908) with 119899 verticessuch that 119899 ge 3we have
log (119899) le 119868 (119866 119860119861119862) le log(119899 (119899 minus 1)2 ) (10)
and log(119899) = 119868(119866 119860119861119862) if and only if G is cycle graph and119868(119866 119860119861119862) = log(119899(119899minus1)2) if and only if G is complete graph
Proof Let a 119896 regular graph G with 119896 ge 2 As G is connectedwith 119899 ge 3 so119868 (119866 119860119861119862)
= minussum radic(2119896 minus 2) 1198962sumradic(2119896 minus 2) 1198962 log( radic(2119896 minus 2) 1198962sumradic(2119896 minus 2) 1198962)= minussum 2119899119896 log( 2119899119896) = minus log(1198991198962 )
(11)
Since 2 le 119896 le 119899 minus 1 we havelog (119899) le 119868 (119866 119860119861119862) le log(119899 (119899 minus 1)2 ) (12)
Theorem 5 For a complete bipartite graph119866 = (119881 119864 119908)with119899 vertices we havelog (119899 minus 1) le 119868 (119866 119860119861119862) le log(lfloor1198992rfloor lceil1198992rceil) (13)
and log(119899 minus 1) = 119868(119866 119860119861119862) if and only if G is star graphand 119868(119866 119860119861119862) = log(lfloor1198992rfloorlceil1198992rceil) if and only if G is completebipartite graph (balanced)
Proof For a complete bipartite graph 119866 = (119881 119864 119908) with 119899vertices and two parts have 119901 and 119902 vertices respectivelyTherefore 119901 + 119902 = 119899 We have
119868 (119866 119860119861119862) = minussum radic(119901 + 119902 minus 2) 119901119902sumradic(119901 + 119902 minus 2) 119901119902
sdot log( radic(119901 + 119902 minus 2) 119901119902sumradic(119901 + 119902 minus 2) 119901119902)
119868 (119866 119860119861119862) = minussum radic(119901 + 119902 minus 2) 119901119902119901119902radic(119901 + 119902 minus 2) 119901119902
sdot log( radic(119901 + 119902 minus 2) 119901119902119901119902radic(119901 + 119902 minus 2) 119901119902) = minussum( 1119901119902)
sdot log(( 1119901119902)) = log (119901119902)
(14)
Thus
log (119899 minus 1) le 119868 (119866 119860119861119862) le log(lfloor1198992rfloor lceil1198992rceil) (15)
Moreover log(119899 minus 1) = 119868(119866 119860119861119862) if and only if 119901 = 1 and119902 = 119899 minus 1 ie G is a star Also 119868(119866 119860119861119862) = log(lfloor1198992rfloorlceil1198992rceil)if and only if 119901 = lfloor1198992rfloor and 119902 = lceil1198992rceil ie G is a completebipartite graph (balanced)
Chemical graph is a graph associated with the chemicalcompound in which atoms are taken as vertices and chemicalbonds are taken as edges In the following theorem we givebounds for theweighted entropy of chemical graphs by takingABC edge weights
Theorem6 Let G be a chemical graph with 119899 vertices then wehave
log (119860119861119862) minus logradic 116 le 119868 (119866 119860119861119862)le log (119860119861119862) minus logradic3 (16)
Proof In a chemical graphG themaximumdegree of a vertexis 4 and the minimum degree of a vertex is 1 so we have
119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sumradic119889119906 + 119889V minus 2119889119906119889Vsdot log(radic119889119906 + 119889V minus 2119889119906119889V )
4 Discrete Dynamics in Nature and Society
= log (119860119861119862) minus 12 (119860119861119862) sumradic119889119906 + 119889V minus 2119889119906119889Vsdot [log(radic119889119906 + 119889V minus 2) minus log (119889119906119889V) )]le log (119860119861119862) minus 12 [log (4 + 4 minus 2) minus log (2)]le log (119860119861119862) minus logradic3
(17)
Similarly
119868 (119866 119860119861119862) ge log (119860119861119862) minus logradic 116 (18)
Therefore
log (119860119861119862) minus logradic 116 le 119868 (119866 119860119861119862)le log (119860119861119862) minus logradic3 (19)
Corollary 7 Let119866 = (119881 119864 119908) be any complete graph of order119899 Then we have
119868 (119866 119860119861119862) le 119899radic119899 minus 22 minus log(radic 119899119899 minus 1) (20)
Proof For any complete graph G of order 119899 we have [14]119860119861119862(119866) le 119899radic(119899 minus 2)2 therefore the result119868 (119866 119860119861119862) le 119899radic119899 minus 22 minus log(radic 119899119899 minus 1) (21)
Corollary 8 Let119866 = (119881 119864 119908) be any tree of order 119899 Then wehave
119868 (119866 119860119861119862) le radic(119899 minus 1) (119899 minus 2) minus log(radic 119899119899 minus 1) (22)
Corollary 9 For a unicyclic graph
119868 (119866 119860119861119862) le radic 119899 (21198992 minus 7119899 minus 19)2 (119899 minus 1) minus log(radic 119899119899 minus 1) (23)
Corollary 10 For a Star graph
119868 (119866 119860119861119862) le radic 119899 (21198992 minus 7119899 minus 19)2 (119899 minus 1) minus log(radic 119899119899 minus 1) (24)
3 Numerical Examples
Dendrimers are man-made nanoscale compounds withunique properties that make them useful to the health and
pharmaceutical industry as both enhancements to existingproducts and as entirely new products Dendrimers areconstructed by the successive addition of layers of branch-ing groups The final generation incorporates the surfacemolecules that give the dendrimers the desired function forpharmaceutical life science chemical electronic and mate-rials applicationsDendrimers fall under the broad heading ofnanotechnology which covers the manipulation of matter inthe size range of 1-100 nanometers (one million nanometersequal one millimeter) to create compounds structures anddevices with a novel predetermined properties
In this section we discuss entropies of four famil-iar classes of dendrimers namely Porphyrin (Figure 1)Propyl ether imine (Figure 2) Zinc-Porphyrin (Figure 3)and Poly(EThyleneAmidoAmine) (Figure 4) Dendrimers Itis important to remark that all dendrimers differ in coresThese dendrimers have been studied extensively in [25ndash28]
Example 1 Let G be the Porphyrin dendrimers then usingthe edge partition given of Porphyrin dendrimers given inTable 1 we get119860119861119862 (119866) = 7726044062119899 minus 7778174591 (25)
Therefore
119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (772604119899 minus 77781)minus 12 (772604119899 minus 77781) 100381610038161003816100381611986411003816100381610038161003816 (radic63 ) log(radic63 )+ 100381610038161003816100381611986421003816100381610038161003816 (radic32 ) log(radic32 ) + 100381610038161003816100381611986431003816100381610038161003816 (radic22 ) log(radic22 )+ 100381610038161003816100381611986441003816100381610038161003816 (radic22 ) log(radic22 ) + 100381610038161003816100381611986451003816100381610038161003816 (23) log(23)+ (100381610038161003816100381611986461003816100381610038161003816 (radic156 ) log(radic156 ))= log (772604119899 minus 77781)minus 12 (772604119899 minus 77781) (2119899) (radic63 ) log(radic63 )+ (24119899) (radic32 ) log(radic32 ) + (10119899 minus 5) (radic22 ) log(radic22 ) + (48119899 minus 6) (radic22 ) log(radic22 ) + (13119899)sdot (23) log(23) + ((8119899) (radic156 ) log(radic156 ))= log (772604119899 minus 77781)minus 12 (772604119899 minus 77781) (minus23308119899 + 2695)
(26)
Discrete Dynamics in Nature and Society 5
Table 1 Edge partition of Porphyrin dendrimers based on degree of end vertices of each edge
(119889119906119889V) (13) (14) (22) (23) (33) (34)Number of edges 2n 24n 10n-5 48n-6 13n 8n
Figure 1 Porphyrin dendrimer
Example 2 Let G be the propyl ether imine dendrimerThenusing the edge partition of G given in Table 2 we get
ABC (G) = 07072119899+1 + 07072119899+4 minus 16970 + 42422119899119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906 log(radic119889119906 + 119889119906 minus 2119889119906119889119906 )
6 Discrete Dynamics in Nature and Society
O O O
OO
O
O
O
O
O
O
O
O
OO
O
O
O
O
OO
O
OO
O
O
O
O
O
O
O
O
OO
O
O O O
O
O
O
OO
O
OO
O
O
O
O
O
O
OO
O O
O
O
O
N
N
N
N
NN
N
N
N
NN
N
NNN
N
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N N
NNN
N
N
NN
N
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2（2
N
O
N
O
N
Figure 2 Propyl ether imine dendrimer
= log (07072119899+1 + 07072119899+4 minus 16970 + 42422119899)minus 12 (07072119899+1 + 07072119899+4 minus 16970 + 42422119899) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))] + [100381610038161003816100381611986421003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986431003816100381610038161003816 ((radic22 ) log(radic22 ))] = log (07072119899+1 + 07072119899+4 minus 16970 + 42422119899)minus 12 (07072119899+1 + 07072119899+4 minus 16970 + 42422119899) [(2119899+1)((radic22 ) log(radic22 ))] + (2119899+4)((radic22 ) log(radic22 )) ]
+ (62119899 minus 6)((radic22 ) log(radic22 )) ]
= minus 1ln (070710678102119899+1 + 070710678102119899+4 minus 1697056275 + 0707106781062119899) (minus024506453602119899+1minus 024506453602119899+4 + 5881548863 minus 0245064536062119899)
(27)
Example 3 For the Zinc-Porphyrin dendrimers G usingedge partition given in Table 3 we get
ABC (G) = 449312119899 minus 22226119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906
sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (449312119899 minus 22226)minus 12 (449312119899 minus 22226) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986421003816100381610038161003816 ((radic22 ) log(radic22 ))]
Discrete Dynamics in Nature and Society 7
N N N
N
N
N
N
N
NN
N
N
N
NN
N
N
NN
N
N
N
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N
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NNN
NNZn
N
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N N N
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N
Figure 3 Zinc-Porphyrin dendrimer
+ [100381610038161003816100381611986431003816100381610038161003816 ((23) log(23))]+ [100381610038161003816100381611986441003816100381610038161003816 ((radic156 ) log(radic156 ))]= log (449312119899 minus 22226) minus 12 (449312119899 minus 22226)sdot [(162119899 minus 4)((radic22 ) log(radic22 ))] + (402119899 minus 16)sdot ((radic22 ) log(radic22 )) ] + (82119899 minus 16) ((23) log(23)) ]+ (4) ((radic156 ) log(radic156 )) ] = ln (44931313072119899
minus 2222681339)minus 1ln (44931313072119899 minus 2222681339) (minus15886094592119899
minus 8096026594)(28)
Example 4 Let G be the graph of Poly(EThyleneAmido-Amine then using edge partition given in Table 4 we get
ABC (G) = 315502119899 minus 13653119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906
8 Discrete Dynamics in Nature and Society
G5
Figure 4 Poly(EThyleneAmidoAmine dendrimer
sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (315502119899 minus 13653)minus 12 (315502119899 minus 13653) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986421003816100381610038161003816 ((radic63 ) log(radic63 )]+ [100381610038161003816100381611986431003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986441003816100381610038161003816 ((radic22 ) log(radic22 ))]= log (449312119899 minus 22226) minus 12 (449312119899 minus 22226)sdot (42119899) ((radic22 ) log(radic22 )) ] + (42119899 minus 2)sdot ((radic22 ) log(radic22 )) ] + (162119899) ((23) log(23)) ]+ (202119899 minus 9)((radic156 ) log(radic156 )) ]
Table 2 Edge partition of propyl ether imine dendrimers based ondegree of end vertices of each edge
(119889119906119889V) (12) (22) (23)Number of edges 2119899+1 2119899+4 62119899 minus 6
= ln (31550257562119899 minus 1365380844)minus 1ln (31550257562119899 minus 1365380844) (minus10464703192119899+ 4497157986)
(29)
Concluding Remarks QSARs represent predictive mod-els got from utilization of statistical instruments corre-lating biological activity (including desirable therapeuticeffect and undesirable side effects) of chemicals (toxi-cantsdrugsenvironmental pollutants) with descriptors illus-trative of molecular structure as well as properties QSARsare being connected inmany disciplines for instance toxicityprediction risk assessment and regulatory decisions leadoptimization and drug discoveryThe atom-bond connectiv-ity index denoted by ABC is a molecular structure descriptorthat has remarkable application in rationalizing the stability
Discrete Dynamics in Nature and Society 9
Table 3 Edge partition of Zinc-Porphyrin dendrimers based on degree of end vertices of each edge
(119889119906119889V) (22) (23) (33) (3 4)Number of edges 162119899-4 402119899 minus 16 82119899 minus 16 4
Table 4 Edge partition of Poly(EThyleneAmidoAmine dendrimers based on degree of end vertices of each edge
(119889119906119889V) (12) (13) (22) (2 3)Number of edges 42119899 42119899 minus 2 162119899 202119899 minus 9of linear and branched alkanes and in the strain energy ofcycloalkanes [25] Weighted entropy is a generalization ofShannonrsquos entropy and is themeasure of information suppliedby a probablistic experiment whose elementary events arecharacterized both by their objective probabilities and bysome qualitative (objective or subjective) weights [29] Itis useful to rank chemicals in quantitative high-throughputscreening experiments [30] and may be used to balancethe amount of information and the degree of homogeneityassociated to a partition of data in classes [31] Weightedentropy also found applications in the coding theory [32] Formore insights about applications of entropy please see [33]In this paper we have studied weighted entropy with atomicbond connectivity edge weights which was an open problemof [34] Our next aim is to work on entropy of weightedgraphs with geometric arithmetic and sum connectivity edgeweights
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
Authors do not have any competing interests
Authorsrsquo Contributions
All authors contributed equally to this paper
Acknowledgments
This work was supported by the Dong-A University researchfund
References
[1] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 2016
[2] W Gao M R Farahani and L Shi ldquoThe forgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterranea vol32 no 1 pp 579ndash585 2016
[3] S Kang Z Iqbal M Ishaq R Sarfraz A Aslam andW NazeerldquoOn eccentricity-based topological indices and polynomials ofphosphorus-containing dendrimersrdquo Symmetry vol 10 no 7 p237 2018
[4] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe degree-based topological indices of dendrimersrdquoChemistryvol 16 no 1 pp 681ndash688 2018
[5] S M Kang W Nazeer M A Zahid A R Nizami A Aslamand M Munir ldquoM-polynomials and topological indices of hex-derived networksrdquo Physics vol 16 no 1 pp 394ndash403 2018
[6] HWiener ldquoStructural determination of paraffinboiling pointsrdquoJournal of the American Chemical Society vol 69 no 1 pp 17ndash20 1947
[7] M Randic ldquoOn characterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23 pp6609ndash6615 1975
[8] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity indexmodeling the enthalpy of formation ofalkanesrdquo Indian Journal of Chemistry vol 37 pp 849ndash855 1998
[9] M Ghorbani and M A Hosseinzadeh ldquoHosseinzadeh Com-puting index of nanostar dendrimers Opto-electronrdquoOptoelec-tronics and Advanced Materials Rapid Communications vol 4no 9 pp 1419ndash1422 2010
[10] W Nazeer A Farooq M Younas M Munir and S Kang ldquoOnMolecular Descriptors of Carbon Nanoconesrdquo Biomoleculesvol 8 no 3 p 92 2018
[11] W Gao M Younas A Farooq A Mahboob and W NazeerldquoM-Polynomials and Degree-Based Topological Indices of theCrystallographic Structure of Moleculesrdquo Biomolecules vol 8no 4 p 107 2018
[12] Y C Kwun M Munir W Nazeer S Rafique and S M KangldquoComputational Analysis of topological indices of two BoronNanotubesrdquo Scientific reports vol 8 no 1 p 14843 2018
[13] W Gao M Younas A Farooq A Virk and W Nazeer ldquoSomeReverse Degree-Based Topological Indices and Polynomials ofDendrimersrdquoMathematics vol 6 no 10 p 214 2018
[14] M Munir W Nazeer S Rafique A R Nizami and S M KangldquoSome computational aspects of boron triangular nanotubesrdquoSymmetry vol 9 no 1 2017
[15] Y C Kwun M Munir W Nazeer S Rafique and S MKang ldquoM-Polynomials and topological indices of V-PhenylenicNanotubes and Nanotorirdquo Scientific Reports vol 7 no 1 2017
[16] C E Shannon The Mathematical Theory of CommunicationThe University of Illinois Press Urbana Ill USA 1949
[17] A Mowshowitz ldquoEntropy and the complexity of graphs II Theinformation content of digraphs and infinite graphsrdquo Bulletin ofMathematical Biology vol 30 pp 225ndash240 1968
[18] A Mowshowitz ldquoEntropy and the complexity of graphs IAn index of the relative complexity of a graphrdquo Bulletin ofMathematical Biology vol 30 pp 175ndash204 1968
[19] J Korner ldquoCoding of an information source having ambiguousalphabet and the entropy of graphsrdquo the 6th Pargue Conferenceon Information Theory Statistical Decision Functions RandomProcesses Pargue Czech Republic pp 411ndash425 1973
10 Discrete Dynamics in Nature and Society
[20] M M Dehmer N N Barbarini K K Varmuza and A AGraber ldquoNovel topological descriptors for analyzing biologicalnetworksrdquo BMC Structural Biology vol 10 article no 18 2010
[21] M Dehmer and A Mowshowitz ldquoA history of graph entropymeasuresrdquo Information Sciences vol 181 no 1 pp 57ndash78 2011
[22] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropybounds for dendrimersrdquo Applied Mathematics and Computa-tion vol 242 pp 462ndash472 2014
[23] S Ji X Li and B Huo ldquoOn reformulated Zagreb indices withrespect to acyclic unicyclic and bicyclic graphsrdquo The Matchvol 72 no 3 pp 723ndash732 2014
[24] K Xu K C Das and S Balachandran ldquoMaximizing theZagreb Indices of (nm)-Graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 72 pp 641ndash6542014
[25] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquo Filomat vol 26 no 4 pp 733ndash738 2012
[26] Y Bashir A AslamM Kamran et al ldquoOn forgotten topologicalindices of some dendrimers structurerdquoMolecules vol 22 no 62017
[27] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe Degree-based Topological Indices of Dendrimersrdquo OpenChemistry 2018
[28] A Aslam Y Bashir M Rafiq F Haider N MuhammadandN Bibi ldquoThree NewOld Vertex-Degree-Based TopologicalIndices of Some Dendrimers Structurerdquo Journal of Biology vol13 no 1 2017
[29] S Guiasu ldquoWeighted entropyrdquoReports onMathematical Physicsvol 2 no 3 pp 165ndash179 1971
[30] K R Shockley ldquoUsing weighted entropy to rank chemicals inquantitative high-throughput screening experimentsrdquo Journalof Biomolecular Screening vol 19 no 3 pp 344ndash353 2014
[31] S Guiasu ldquoGrouping data by using the weighted entropyrdquoJournal of Statistical Planning and Inference vol 15 no 1 pp63ndash69 1986
[32] A Clim ldquoWeighted entropy with applicationrdquo Analele Univer-sitatii Bucuresti Matematica pp 223ndash231 2008
[33] M Ghorbani M Dehmer and S Zangi ldquoGraph operationsbased on using distance-based graph entropiesrdquo Applied Math-ematics and Computation vol 333 pp 547ndash555 2018
[34] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropy ofweighted graphs with Randic weightsrdquo Entropy vol 17 no 6 pp3710ndash3723 2015
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2 Discrete Dynamics in Nature and Society
biology chemistry and computer science Therefore it is notsurprising to realize that ldquograph entropyrdquo has been defined invarious ways Another classic example is the introduction ofKorners entropy [19]
Different graph invariances have been used to developimage entropy measures such as eigenvalue and connectivityinformation [20] distance-based graph entropy [21] anddegree-based graph entropy [22]
We have different applications of graph entropy in com-munications and economics We use the concept of graphentropy as a weighted graph just as Dehmer [20] who solvedthe problem of weighted chemical graph entropy by usinga special information functional Some degree-based indicesare characterized by investigating the extremes of the entropyof certain class of graphs [23 24]
In [25] Chen et al introduced the concept of graphentropy for special weighted graphs by using Randic edgeweights and proved extremal properties of graph entropyfor some elementary families of graphs Our aim is tosolve problem suggested by Chen et al in [25] In thispaper we study graph entropy by taking Atomic bondconnectivity edge weights and prove some extermal proper-ties of graph entropy for special families of graphs such asconnected graphs regular graphs complete bipartite graphschemical graphs tree unicyclic graphs and star graphsMoreover we compute graph entropy of different dendrimerstructures
2 Main Results
Let us have a graph where G and V119894 are its vertices and thedegree of V119894 is denoted by 119889119894 For an edge V119894V119895 we have
119901119894119895 = 119908 (V119894V119895)sum119889119894119895=1 119908(V119894V119895) (2)
where119908(V119894V119895) is the weight V119894V119895 and119908(V119894V119895)gt 0Theweightedentropy is defined as
119867(V119894) = minus 119889119894sum119895=1
119901119894119895 log (119901119894119895) (3)
Inspired by this method we have introduced the definitionof the entropy of the edge-weighted graph which can alsobe interpreted as multiple graphs For edge weight graphsG = (V E w) where V E and w denote the vertex sets ofG edge sets and edge weights (sometimes also called costs)In this article we always assume that the edge weights arepositive
Definition 1 Let 119866 = (119881 119864 119908) be an edge weighted then theentropy of G is
119868 (119866 119908) = minus sum119906Visin119864
119901119906V log (119901119906V) (4)
where 119901119906V = 119908(119906V)sum119906Visin119864119908(119906V)Theorem 2 For a connected graph G with 119899 vertices for 119899 ge 3we have
log (119860119861119862) + logradic 1119899 minus 2 le 119868 (119866 119860119861119862)le log (119860119861119862) minus log 1119899 minus 1
(5)
Proof For a simple connected graphof order 119899 themaximumdegree for a vertex is n-1 and minimum degree is 1 With anyedge uv the minimum possible degrees of u and v are 1 and2 respectively and maximum possible degrees of u and v aren-1 and n-1 so we have
119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sumradic119889119906 + 119889119906 minus 2119889119906119889119906sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (119860119861119862)minus 12 (119860119861119862) sumradic119889119906 + 119889119906 minus 2119889119906119889119906sdot [log(radic119889119906 + 119889V minus 2) minus log (119889119906119889V) )]le log (119860119861119862) minus 12 [log (1) minus log (119899 minus 1)2]= log (119860119861119862) minus logradic 1(119899 minus 1)2 le log (119860119861119862)minus log 1119899 minus 1
I (GABC) ge log (119860119861119862) minus 12 (119860119861119862) sumradic119889119906 + 119889119906 minus 2119889119906119889119906sdot [log (2119899 minus 4) minus log (2)] = log (119860119861119862)minus 12 log (119899 minus 2) = log (119860119861119862) + logradic 1119899 minus 2= log (119860119861119862) + logradic 1119899 minus 2
(6)
Corollary 3 Let G be a graph with 119899 vertices Let 120575 119886119899119889 Δ bethe minimum degree and maximum degree of G respectivelyThen we have
Discrete Dynamics in Nature and Society 3
log (119860119861119862) + log(radic 1205752Δ minus 2) le I (GABC)le log (119860119861119862) minus 12 log( Δradic2120575 minus 2)
(7)
Proof
I (GABC) le log (119860119861119862)minus 12 (119860119861119862) sumradic119889119906 + 119889V minus 2119889119906119889V [log (2120575 minus 2) minus log (Δ2)]= log (119860119861119862) minus 12 log( Δradic2120575 minus 2)
(8)
Also
I (GABC) ge log (119860119861119862) minus 12 [log (2Δ minus 2) minus log (1205752)]= log (119860119861119862) + log(radic 1205752Δ minus 2) (9)
Theorem 4 For a regular graph 119866 = (119881 119864 119908) with 119899 verticessuch that 119899 ge 3we have
log (119899) le 119868 (119866 119860119861119862) le log(119899 (119899 minus 1)2 ) (10)
and log(119899) = 119868(119866 119860119861119862) if and only if G is cycle graph and119868(119866 119860119861119862) = log(119899(119899minus1)2) if and only if G is complete graph
Proof Let a 119896 regular graph G with 119896 ge 2 As G is connectedwith 119899 ge 3 so119868 (119866 119860119861119862)
= minussum radic(2119896 minus 2) 1198962sumradic(2119896 minus 2) 1198962 log( radic(2119896 minus 2) 1198962sumradic(2119896 minus 2) 1198962)= minussum 2119899119896 log( 2119899119896) = minus log(1198991198962 )
(11)
Since 2 le 119896 le 119899 minus 1 we havelog (119899) le 119868 (119866 119860119861119862) le log(119899 (119899 minus 1)2 ) (12)
Theorem 5 For a complete bipartite graph119866 = (119881 119864 119908)with119899 vertices we havelog (119899 minus 1) le 119868 (119866 119860119861119862) le log(lfloor1198992rfloor lceil1198992rceil) (13)
and log(119899 minus 1) = 119868(119866 119860119861119862) if and only if G is star graphand 119868(119866 119860119861119862) = log(lfloor1198992rfloorlceil1198992rceil) if and only if G is completebipartite graph (balanced)
Proof For a complete bipartite graph 119866 = (119881 119864 119908) with 119899vertices and two parts have 119901 and 119902 vertices respectivelyTherefore 119901 + 119902 = 119899 We have
119868 (119866 119860119861119862) = minussum radic(119901 + 119902 minus 2) 119901119902sumradic(119901 + 119902 minus 2) 119901119902
sdot log( radic(119901 + 119902 minus 2) 119901119902sumradic(119901 + 119902 minus 2) 119901119902)
119868 (119866 119860119861119862) = minussum radic(119901 + 119902 minus 2) 119901119902119901119902radic(119901 + 119902 minus 2) 119901119902
sdot log( radic(119901 + 119902 minus 2) 119901119902119901119902radic(119901 + 119902 minus 2) 119901119902) = minussum( 1119901119902)
sdot log(( 1119901119902)) = log (119901119902)
(14)
Thus
log (119899 minus 1) le 119868 (119866 119860119861119862) le log(lfloor1198992rfloor lceil1198992rceil) (15)
Moreover log(119899 minus 1) = 119868(119866 119860119861119862) if and only if 119901 = 1 and119902 = 119899 minus 1 ie G is a star Also 119868(119866 119860119861119862) = log(lfloor1198992rfloorlceil1198992rceil)if and only if 119901 = lfloor1198992rfloor and 119902 = lceil1198992rceil ie G is a completebipartite graph (balanced)
Chemical graph is a graph associated with the chemicalcompound in which atoms are taken as vertices and chemicalbonds are taken as edges In the following theorem we givebounds for theweighted entropy of chemical graphs by takingABC edge weights
Theorem6 Let G be a chemical graph with 119899 vertices then wehave
log (119860119861119862) minus logradic 116 le 119868 (119866 119860119861119862)le log (119860119861119862) minus logradic3 (16)
Proof In a chemical graphG themaximumdegree of a vertexis 4 and the minimum degree of a vertex is 1 so we have
119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sumradic119889119906 + 119889V minus 2119889119906119889Vsdot log(radic119889119906 + 119889V minus 2119889119906119889V )
4 Discrete Dynamics in Nature and Society
= log (119860119861119862) minus 12 (119860119861119862) sumradic119889119906 + 119889V minus 2119889119906119889Vsdot [log(radic119889119906 + 119889V minus 2) minus log (119889119906119889V) )]le log (119860119861119862) minus 12 [log (4 + 4 minus 2) minus log (2)]le log (119860119861119862) minus logradic3
(17)
Similarly
119868 (119866 119860119861119862) ge log (119860119861119862) minus logradic 116 (18)
Therefore
log (119860119861119862) minus logradic 116 le 119868 (119866 119860119861119862)le log (119860119861119862) minus logradic3 (19)
Corollary 7 Let119866 = (119881 119864 119908) be any complete graph of order119899 Then we have
119868 (119866 119860119861119862) le 119899radic119899 minus 22 minus log(radic 119899119899 minus 1) (20)
Proof For any complete graph G of order 119899 we have [14]119860119861119862(119866) le 119899radic(119899 minus 2)2 therefore the result119868 (119866 119860119861119862) le 119899radic119899 minus 22 minus log(radic 119899119899 minus 1) (21)
Corollary 8 Let119866 = (119881 119864 119908) be any tree of order 119899 Then wehave
119868 (119866 119860119861119862) le radic(119899 minus 1) (119899 minus 2) minus log(radic 119899119899 minus 1) (22)
Corollary 9 For a unicyclic graph
119868 (119866 119860119861119862) le radic 119899 (21198992 minus 7119899 minus 19)2 (119899 minus 1) minus log(radic 119899119899 minus 1) (23)
Corollary 10 For a Star graph
119868 (119866 119860119861119862) le radic 119899 (21198992 minus 7119899 minus 19)2 (119899 minus 1) minus log(radic 119899119899 minus 1) (24)
3 Numerical Examples
Dendrimers are man-made nanoscale compounds withunique properties that make them useful to the health and
pharmaceutical industry as both enhancements to existingproducts and as entirely new products Dendrimers areconstructed by the successive addition of layers of branch-ing groups The final generation incorporates the surfacemolecules that give the dendrimers the desired function forpharmaceutical life science chemical electronic and mate-rials applicationsDendrimers fall under the broad heading ofnanotechnology which covers the manipulation of matter inthe size range of 1-100 nanometers (one million nanometersequal one millimeter) to create compounds structures anddevices with a novel predetermined properties
In this section we discuss entropies of four famil-iar classes of dendrimers namely Porphyrin (Figure 1)Propyl ether imine (Figure 2) Zinc-Porphyrin (Figure 3)and Poly(EThyleneAmidoAmine) (Figure 4) Dendrimers Itis important to remark that all dendrimers differ in coresThese dendrimers have been studied extensively in [25ndash28]
Example 1 Let G be the Porphyrin dendrimers then usingthe edge partition given of Porphyrin dendrimers given inTable 1 we get119860119861119862 (119866) = 7726044062119899 minus 7778174591 (25)
Therefore
119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (772604119899 minus 77781)minus 12 (772604119899 minus 77781) 100381610038161003816100381611986411003816100381610038161003816 (radic63 ) log(radic63 )+ 100381610038161003816100381611986421003816100381610038161003816 (radic32 ) log(radic32 ) + 100381610038161003816100381611986431003816100381610038161003816 (radic22 ) log(radic22 )+ 100381610038161003816100381611986441003816100381610038161003816 (radic22 ) log(radic22 ) + 100381610038161003816100381611986451003816100381610038161003816 (23) log(23)+ (100381610038161003816100381611986461003816100381610038161003816 (radic156 ) log(radic156 ))= log (772604119899 minus 77781)minus 12 (772604119899 minus 77781) (2119899) (radic63 ) log(radic63 )+ (24119899) (radic32 ) log(radic32 ) + (10119899 minus 5) (radic22 ) log(radic22 ) + (48119899 minus 6) (radic22 ) log(radic22 ) + (13119899)sdot (23) log(23) + ((8119899) (radic156 ) log(radic156 ))= log (772604119899 minus 77781)minus 12 (772604119899 minus 77781) (minus23308119899 + 2695)
(26)
Discrete Dynamics in Nature and Society 5
Table 1 Edge partition of Porphyrin dendrimers based on degree of end vertices of each edge
(119889119906119889V) (13) (14) (22) (23) (33) (34)Number of edges 2n 24n 10n-5 48n-6 13n 8n
Figure 1 Porphyrin dendrimer
Example 2 Let G be the propyl ether imine dendrimerThenusing the edge partition of G given in Table 2 we get
ABC (G) = 07072119899+1 + 07072119899+4 minus 16970 + 42422119899119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906 log(radic119889119906 + 119889119906 minus 2119889119906119889119906 )
6 Discrete Dynamics in Nature and Society
O O O
OO
O
O
O
O
O
O
O
O
OO
O
O
O
O
OO
O
OO
O
O
O
O
O
O
O
O
OO
O
O O O
O
O
O
OO
O
OO
O
O
O
O
O
O
OO
O O
O
O
O
N
N
N
N
NN
N
N
N
NN
N
NNN
N
N
N N
NNN
N
N
NN
N
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2（2
N
O
N
O
N
Figure 2 Propyl ether imine dendrimer
= log (07072119899+1 + 07072119899+4 minus 16970 + 42422119899)minus 12 (07072119899+1 + 07072119899+4 minus 16970 + 42422119899) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))] + [100381610038161003816100381611986421003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986431003816100381610038161003816 ((radic22 ) log(radic22 ))] = log (07072119899+1 + 07072119899+4 minus 16970 + 42422119899)minus 12 (07072119899+1 + 07072119899+4 minus 16970 + 42422119899) [(2119899+1)((radic22 ) log(radic22 ))] + (2119899+4)((radic22 ) log(radic22 )) ]
+ (62119899 minus 6)((radic22 ) log(radic22 )) ]
= minus 1ln (070710678102119899+1 + 070710678102119899+4 minus 1697056275 + 0707106781062119899) (minus024506453602119899+1minus 024506453602119899+4 + 5881548863 minus 0245064536062119899)
(27)
Example 3 For the Zinc-Porphyrin dendrimers G usingedge partition given in Table 3 we get
ABC (G) = 449312119899 minus 22226119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906
sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (449312119899 minus 22226)minus 12 (449312119899 minus 22226) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986421003816100381610038161003816 ((radic22 ) log(radic22 ))]
Discrete Dynamics in Nature and Society 7
N N N
N
N
N
N
N
NN
N
N
N
NN
N
N
NN
N
N
N
N
N
N
N
N
NN
NNN
NNZn
N
NN
N
N
N
N
N
N
N
NN
N N N
NN
N
N
N
N
N
N
NN
N
NN
N
N
Figure 3 Zinc-Porphyrin dendrimer
+ [100381610038161003816100381611986431003816100381610038161003816 ((23) log(23))]+ [100381610038161003816100381611986441003816100381610038161003816 ((radic156 ) log(radic156 ))]= log (449312119899 minus 22226) minus 12 (449312119899 minus 22226)sdot [(162119899 minus 4)((radic22 ) log(radic22 ))] + (402119899 minus 16)sdot ((radic22 ) log(radic22 )) ] + (82119899 minus 16) ((23) log(23)) ]+ (4) ((radic156 ) log(radic156 )) ] = ln (44931313072119899
minus 2222681339)minus 1ln (44931313072119899 minus 2222681339) (minus15886094592119899
minus 8096026594)(28)
Example 4 Let G be the graph of Poly(EThyleneAmido-Amine then using edge partition given in Table 4 we get
ABC (G) = 315502119899 minus 13653119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906
8 Discrete Dynamics in Nature and Society
G5
Figure 4 Poly(EThyleneAmidoAmine dendrimer
sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (315502119899 minus 13653)minus 12 (315502119899 minus 13653) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986421003816100381610038161003816 ((radic63 ) log(radic63 )]+ [100381610038161003816100381611986431003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986441003816100381610038161003816 ((radic22 ) log(radic22 ))]= log (449312119899 minus 22226) minus 12 (449312119899 minus 22226)sdot (42119899) ((radic22 ) log(radic22 )) ] + (42119899 minus 2)sdot ((radic22 ) log(radic22 )) ] + (162119899) ((23) log(23)) ]+ (202119899 minus 9)((radic156 ) log(radic156 )) ]
Table 2 Edge partition of propyl ether imine dendrimers based ondegree of end vertices of each edge
(119889119906119889V) (12) (22) (23)Number of edges 2119899+1 2119899+4 62119899 minus 6
= ln (31550257562119899 minus 1365380844)minus 1ln (31550257562119899 minus 1365380844) (minus10464703192119899+ 4497157986)
(29)
Concluding Remarks QSARs represent predictive mod-els got from utilization of statistical instruments corre-lating biological activity (including desirable therapeuticeffect and undesirable side effects) of chemicals (toxi-cantsdrugsenvironmental pollutants) with descriptors illus-trative of molecular structure as well as properties QSARsare being connected inmany disciplines for instance toxicityprediction risk assessment and regulatory decisions leadoptimization and drug discoveryThe atom-bond connectiv-ity index denoted by ABC is a molecular structure descriptorthat has remarkable application in rationalizing the stability
Discrete Dynamics in Nature and Society 9
Table 3 Edge partition of Zinc-Porphyrin dendrimers based on degree of end vertices of each edge
(119889119906119889V) (22) (23) (33) (3 4)Number of edges 162119899-4 402119899 minus 16 82119899 minus 16 4
Table 4 Edge partition of Poly(EThyleneAmidoAmine dendrimers based on degree of end vertices of each edge
(119889119906119889V) (12) (13) (22) (2 3)Number of edges 42119899 42119899 minus 2 162119899 202119899 minus 9of linear and branched alkanes and in the strain energy ofcycloalkanes [25] Weighted entropy is a generalization ofShannonrsquos entropy and is themeasure of information suppliedby a probablistic experiment whose elementary events arecharacterized both by their objective probabilities and bysome qualitative (objective or subjective) weights [29] Itis useful to rank chemicals in quantitative high-throughputscreening experiments [30] and may be used to balancethe amount of information and the degree of homogeneityassociated to a partition of data in classes [31] Weightedentropy also found applications in the coding theory [32] Formore insights about applications of entropy please see [33]In this paper we have studied weighted entropy with atomicbond connectivity edge weights which was an open problemof [34] Our next aim is to work on entropy of weightedgraphs with geometric arithmetic and sum connectivity edgeweights
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
Authors do not have any competing interests
Authorsrsquo Contributions
All authors contributed equally to this paper
Acknowledgments
This work was supported by the Dong-A University researchfund
References
[1] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 2016
[2] W Gao M R Farahani and L Shi ldquoThe forgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterranea vol32 no 1 pp 579ndash585 2016
[3] S Kang Z Iqbal M Ishaq R Sarfraz A Aslam andW NazeerldquoOn eccentricity-based topological indices and polynomials ofphosphorus-containing dendrimersrdquo Symmetry vol 10 no 7 p237 2018
[4] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe degree-based topological indices of dendrimersrdquoChemistryvol 16 no 1 pp 681ndash688 2018
[5] S M Kang W Nazeer M A Zahid A R Nizami A Aslamand M Munir ldquoM-polynomials and topological indices of hex-derived networksrdquo Physics vol 16 no 1 pp 394ndash403 2018
[6] HWiener ldquoStructural determination of paraffinboiling pointsrdquoJournal of the American Chemical Society vol 69 no 1 pp 17ndash20 1947
[7] M Randic ldquoOn characterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23 pp6609ndash6615 1975
[8] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity indexmodeling the enthalpy of formation ofalkanesrdquo Indian Journal of Chemistry vol 37 pp 849ndash855 1998
[9] M Ghorbani and M A Hosseinzadeh ldquoHosseinzadeh Com-puting index of nanostar dendrimers Opto-electronrdquoOptoelec-tronics and Advanced Materials Rapid Communications vol 4no 9 pp 1419ndash1422 2010
[10] W Nazeer A Farooq M Younas M Munir and S Kang ldquoOnMolecular Descriptors of Carbon Nanoconesrdquo Biomoleculesvol 8 no 3 p 92 2018
[11] W Gao M Younas A Farooq A Mahboob and W NazeerldquoM-Polynomials and Degree-Based Topological Indices of theCrystallographic Structure of Moleculesrdquo Biomolecules vol 8no 4 p 107 2018
[12] Y C Kwun M Munir W Nazeer S Rafique and S M KangldquoComputational Analysis of topological indices of two BoronNanotubesrdquo Scientific reports vol 8 no 1 p 14843 2018
[13] W Gao M Younas A Farooq A Virk and W Nazeer ldquoSomeReverse Degree-Based Topological Indices and Polynomials ofDendrimersrdquoMathematics vol 6 no 10 p 214 2018
[14] M Munir W Nazeer S Rafique A R Nizami and S M KangldquoSome computational aspects of boron triangular nanotubesrdquoSymmetry vol 9 no 1 2017
[15] Y C Kwun M Munir W Nazeer S Rafique and S MKang ldquoM-Polynomials and topological indices of V-PhenylenicNanotubes and Nanotorirdquo Scientific Reports vol 7 no 1 2017
[16] C E Shannon The Mathematical Theory of CommunicationThe University of Illinois Press Urbana Ill USA 1949
[17] A Mowshowitz ldquoEntropy and the complexity of graphs II Theinformation content of digraphs and infinite graphsrdquo Bulletin ofMathematical Biology vol 30 pp 225ndash240 1968
[18] A Mowshowitz ldquoEntropy and the complexity of graphs IAn index of the relative complexity of a graphrdquo Bulletin ofMathematical Biology vol 30 pp 175ndash204 1968
[19] J Korner ldquoCoding of an information source having ambiguousalphabet and the entropy of graphsrdquo the 6th Pargue Conferenceon Information Theory Statistical Decision Functions RandomProcesses Pargue Czech Republic pp 411ndash425 1973
10 Discrete Dynamics in Nature and Society
[20] M M Dehmer N N Barbarini K K Varmuza and A AGraber ldquoNovel topological descriptors for analyzing biologicalnetworksrdquo BMC Structural Biology vol 10 article no 18 2010
[21] M Dehmer and A Mowshowitz ldquoA history of graph entropymeasuresrdquo Information Sciences vol 181 no 1 pp 57ndash78 2011
[22] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropybounds for dendrimersrdquo Applied Mathematics and Computa-tion vol 242 pp 462ndash472 2014
[23] S Ji X Li and B Huo ldquoOn reformulated Zagreb indices withrespect to acyclic unicyclic and bicyclic graphsrdquo The Matchvol 72 no 3 pp 723ndash732 2014
[24] K Xu K C Das and S Balachandran ldquoMaximizing theZagreb Indices of (nm)-Graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 72 pp 641ndash6542014
[25] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquo Filomat vol 26 no 4 pp 733ndash738 2012
[26] Y Bashir A AslamM Kamran et al ldquoOn forgotten topologicalindices of some dendrimers structurerdquoMolecules vol 22 no 62017
[27] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe Degree-based Topological Indices of Dendrimersrdquo OpenChemistry 2018
[28] A Aslam Y Bashir M Rafiq F Haider N MuhammadandN Bibi ldquoThree NewOld Vertex-Degree-Based TopologicalIndices of Some Dendrimers Structurerdquo Journal of Biology vol13 no 1 2017
[29] S Guiasu ldquoWeighted entropyrdquoReports onMathematical Physicsvol 2 no 3 pp 165ndash179 1971
[30] K R Shockley ldquoUsing weighted entropy to rank chemicals inquantitative high-throughput screening experimentsrdquo Journalof Biomolecular Screening vol 19 no 3 pp 344ndash353 2014
[31] S Guiasu ldquoGrouping data by using the weighted entropyrdquoJournal of Statistical Planning and Inference vol 15 no 1 pp63ndash69 1986
[32] A Clim ldquoWeighted entropy with applicationrdquo Analele Univer-sitatii Bucuresti Matematica pp 223ndash231 2008
[33] M Ghorbani M Dehmer and S Zangi ldquoGraph operationsbased on using distance-based graph entropiesrdquo Applied Math-ematics and Computation vol 333 pp 547ndash555 2018
[34] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropy ofweighted graphs with Randic weightsrdquo Entropy vol 17 no 6 pp3710ndash3723 2015
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Discrete Dynamics in Nature and Society 3
log (119860119861119862) + log(radic 1205752Δ minus 2) le I (GABC)le log (119860119861119862) minus 12 log( Δradic2120575 minus 2)
(7)
Proof
I (GABC) le log (119860119861119862)minus 12 (119860119861119862) sumradic119889119906 + 119889V minus 2119889119906119889V [log (2120575 minus 2) minus log (Δ2)]= log (119860119861119862) minus 12 log( Δradic2120575 minus 2)
(8)
Also
I (GABC) ge log (119860119861119862) minus 12 [log (2Δ minus 2) minus log (1205752)]= log (119860119861119862) + log(radic 1205752Δ minus 2) (9)
Theorem 4 For a regular graph 119866 = (119881 119864 119908) with 119899 verticessuch that 119899 ge 3we have
log (119899) le 119868 (119866 119860119861119862) le log(119899 (119899 minus 1)2 ) (10)
and log(119899) = 119868(119866 119860119861119862) if and only if G is cycle graph and119868(119866 119860119861119862) = log(119899(119899minus1)2) if and only if G is complete graph
Proof Let a 119896 regular graph G with 119896 ge 2 As G is connectedwith 119899 ge 3 so119868 (119866 119860119861119862)
= minussum radic(2119896 minus 2) 1198962sumradic(2119896 minus 2) 1198962 log( radic(2119896 minus 2) 1198962sumradic(2119896 minus 2) 1198962)= minussum 2119899119896 log( 2119899119896) = minus log(1198991198962 )
(11)
Since 2 le 119896 le 119899 minus 1 we havelog (119899) le 119868 (119866 119860119861119862) le log(119899 (119899 minus 1)2 ) (12)
Theorem 5 For a complete bipartite graph119866 = (119881 119864 119908)with119899 vertices we havelog (119899 minus 1) le 119868 (119866 119860119861119862) le log(lfloor1198992rfloor lceil1198992rceil) (13)
and log(119899 minus 1) = 119868(119866 119860119861119862) if and only if G is star graphand 119868(119866 119860119861119862) = log(lfloor1198992rfloorlceil1198992rceil) if and only if G is completebipartite graph (balanced)
Proof For a complete bipartite graph 119866 = (119881 119864 119908) with 119899vertices and two parts have 119901 and 119902 vertices respectivelyTherefore 119901 + 119902 = 119899 We have
119868 (119866 119860119861119862) = minussum radic(119901 + 119902 minus 2) 119901119902sumradic(119901 + 119902 minus 2) 119901119902
sdot log( radic(119901 + 119902 minus 2) 119901119902sumradic(119901 + 119902 minus 2) 119901119902)
119868 (119866 119860119861119862) = minussum radic(119901 + 119902 minus 2) 119901119902119901119902radic(119901 + 119902 minus 2) 119901119902
sdot log( radic(119901 + 119902 minus 2) 119901119902119901119902radic(119901 + 119902 minus 2) 119901119902) = minussum( 1119901119902)
sdot log(( 1119901119902)) = log (119901119902)
(14)
Thus
log (119899 minus 1) le 119868 (119866 119860119861119862) le log(lfloor1198992rfloor lceil1198992rceil) (15)
Moreover log(119899 minus 1) = 119868(119866 119860119861119862) if and only if 119901 = 1 and119902 = 119899 minus 1 ie G is a star Also 119868(119866 119860119861119862) = log(lfloor1198992rfloorlceil1198992rceil)if and only if 119901 = lfloor1198992rfloor and 119902 = lceil1198992rceil ie G is a completebipartite graph (balanced)
Chemical graph is a graph associated with the chemicalcompound in which atoms are taken as vertices and chemicalbonds are taken as edges In the following theorem we givebounds for theweighted entropy of chemical graphs by takingABC edge weights
Theorem6 Let G be a chemical graph with 119899 vertices then wehave
log (119860119861119862) minus logradic 116 le 119868 (119866 119860119861119862)le log (119860119861119862) minus logradic3 (16)
Proof In a chemical graphG themaximumdegree of a vertexis 4 and the minimum degree of a vertex is 1 so we have
119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sumradic119889119906 + 119889V minus 2119889119906119889Vsdot log(radic119889119906 + 119889V minus 2119889119906119889V )
4 Discrete Dynamics in Nature and Society
= log (119860119861119862) minus 12 (119860119861119862) sumradic119889119906 + 119889V minus 2119889119906119889Vsdot [log(radic119889119906 + 119889V minus 2) minus log (119889119906119889V) )]le log (119860119861119862) minus 12 [log (4 + 4 minus 2) minus log (2)]le log (119860119861119862) minus logradic3
(17)
Similarly
119868 (119866 119860119861119862) ge log (119860119861119862) minus logradic 116 (18)
Therefore
log (119860119861119862) minus logradic 116 le 119868 (119866 119860119861119862)le log (119860119861119862) minus logradic3 (19)
Corollary 7 Let119866 = (119881 119864 119908) be any complete graph of order119899 Then we have
119868 (119866 119860119861119862) le 119899radic119899 minus 22 minus log(radic 119899119899 minus 1) (20)
Proof For any complete graph G of order 119899 we have [14]119860119861119862(119866) le 119899radic(119899 minus 2)2 therefore the result119868 (119866 119860119861119862) le 119899radic119899 minus 22 minus log(radic 119899119899 minus 1) (21)
Corollary 8 Let119866 = (119881 119864 119908) be any tree of order 119899 Then wehave
119868 (119866 119860119861119862) le radic(119899 minus 1) (119899 minus 2) minus log(radic 119899119899 minus 1) (22)
Corollary 9 For a unicyclic graph
119868 (119866 119860119861119862) le radic 119899 (21198992 minus 7119899 minus 19)2 (119899 minus 1) minus log(radic 119899119899 minus 1) (23)
Corollary 10 For a Star graph
119868 (119866 119860119861119862) le radic 119899 (21198992 minus 7119899 minus 19)2 (119899 minus 1) minus log(radic 119899119899 minus 1) (24)
3 Numerical Examples
Dendrimers are man-made nanoscale compounds withunique properties that make them useful to the health and
pharmaceutical industry as both enhancements to existingproducts and as entirely new products Dendrimers areconstructed by the successive addition of layers of branch-ing groups The final generation incorporates the surfacemolecules that give the dendrimers the desired function forpharmaceutical life science chemical electronic and mate-rials applicationsDendrimers fall under the broad heading ofnanotechnology which covers the manipulation of matter inthe size range of 1-100 nanometers (one million nanometersequal one millimeter) to create compounds structures anddevices with a novel predetermined properties
In this section we discuss entropies of four famil-iar classes of dendrimers namely Porphyrin (Figure 1)Propyl ether imine (Figure 2) Zinc-Porphyrin (Figure 3)and Poly(EThyleneAmidoAmine) (Figure 4) Dendrimers Itis important to remark that all dendrimers differ in coresThese dendrimers have been studied extensively in [25ndash28]
Example 1 Let G be the Porphyrin dendrimers then usingthe edge partition given of Porphyrin dendrimers given inTable 1 we get119860119861119862 (119866) = 7726044062119899 minus 7778174591 (25)
Therefore
119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (772604119899 minus 77781)minus 12 (772604119899 minus 77781) 100381610038161003816100381611986411003816100381610038161003816 (radic63 ) log(radic63 )+ 100381610038161003816100381611986421003816100381610038161003816 (radic32 ) log(radic32 ) + 100381610038161003816100381611986431003816100381610038161003816 (radic22 ) log(radic22 )+ 100381610038161003816100381611986441003816100381610038161003816 (radic22 ) log(radic22 ) + 100381610038161003816100381611986451003816100381610038161003816 (23) log(23)+ (100381610038161003816100381611986461003816100381610038161003816 (radic156 ) log(radic156 ))= log (772604119899 minus 77781)minus 12 (772604119899 minus 77781) (2119899) (radic63 ) log(radic63 )+ (24119899) (radic32 ) log(radic32 ) + (10119899 minus 5) (radic22 ) log(radic22 ) + (48119899 minus 6) (radic22 ) log(radic22 ) + (13119899)sdot (23) log(23) + ((8119899) (radic156 ) log(radic156 ))= log (772604119899 minus 77781)minus 12 (772604119899 minus 77781) (minus23308119899 + 2695)
(26)
Discrete Dynamics in Nature and Society 5
Table 1 Edge partition of Porphyrin dendrimers based on degree of end vertices of each edge
(119889119906119889V) (13) (14) (22) (23) (33) (34)Number of edges 2n 24n 10n-5 48n-6 13n 8n
Figure 1 Porphyrin dendrimer
Example 2 Let G be the propyl ether imine dendrimerThenusing the edge partition of G given in Table 2 we get
ABC (G) = 07072119899+1 + 07072119899+4 minus 16970 + 42422119899119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906 log(radic119889119906 + 119889119906 minus 2119889119906119889119906 )
6 Discrete Dynamics in Nature and Society
O O O
OO
O
O
O
O
O
O
O
O
OO
O
O
O
O
OO
O
OO
O
O
O
O
O
O
O
O
OO
O
O O O
O
O
O
OO
O
OO
O
O
O
O
O
O
OO
O O
O
O
O
N
N
N
N
NN
N
N
N
NN
N
NNN
N
N
N N
NNN
N
N
NN
N
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2（2
N
O
N
O
N
Figure 2 Propyl ether imine dendrimer
= log (07072119899+1 + 07072119899+4 minus 16970 + 42422119899)minus 12 (07072119899+1 + 07072119899+4 minus 16970 + 42422119899) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))] + [100381610038161003816100381611986421003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986431003816100381610038161003816 ((radic22 ) log(radic22 ))] = log (07072119899+1 + 07072119899+4 minus 16970 + 42422119899)minus 12 (07072119899+1 + 07072119899+4 minus 16970 + 42422119899) [(2119899+1)((radic22 ) log(radic22 ))] + (2119899+4)((radic22 ) log(radic22 )) ]
+ (62119899 minus 6)((radic22 ) log(radic22 )) ]
= minus 1ln (070710678102119899+1 + 070710678102119899+4 minus 1697056275 + 0707106781062119899) (minus024506453602119899+1minus 024506453602119899+4 + 5881548863 minus 0245064536062119899)
(27)
Example 3 For the Zinc-Porphyrin dendrimers G usingedge partition given in Table 3 we get
ABC (G) = 449312119899 minus 22226119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906
sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (449312119899 minus 22226)minus 12 (449312119899 minus 22226) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986421003816100381610038161003816 ((radic22 ) log(radic22 ))]
Discrete Dynamics in Nature and Society 7
N N N
N
N
N
N
N
NN
N
N
N
NN
N
N
NN
N
N
N
N
N
N
N
N
NN
NNN
NNZn
N
NN
N
N
N
N
N
N
N
NN
N N N
NN
N
N
N
N
N
N
NN
N
NN
N
N
Figure 3 Zinc-Porphyrin dendrimer
+ [100381610038161003816100381611986431003816100381610038161003816 ((23) log(23))]+ [100381610038161003816100381611986441003816100381610038161003816 ((radic156 ) log(radic156 ))]= log (449312119899 minus 22226) minus 12 (449312119899 minus 22226)sdot [(162119899 minus 4)((radic22 ) log(radic22 ))] + (402119899 minus 16)sdot ((radic22 ) log(radic22 )) ] + (82119899 minus 16) ((23) log(23)) ]+ (4) ((radic156 ) log(radic156 )) ] = ln (44931313072119899
minus 2222681339)minus 1ln (44931313072119899 minus 2222681339) (minus15886094592119899
minus 8096026594)(28)
Example 4 Let G be the graph of Poly(EThyleneAmido-Amine then using edge partition given in Table 4 we get
ABC (G) = 315502119899 minus 13653119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906
8 Discrete Dynamics in Nature and Society
G5
Figure 4 Poly(EThyleneAmidoAmine dendrimer
sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (315502119899 minus 13653)minus 12 (315502119899 minus 13653) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986421003816100381610038161003816 ((radic63 ) log(radic63 )]+ [100381610038161003816100381611986431003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986441003816100381610038161003816 ((radic22 ) log(radic22 ))]= log (449312119899 minus 22226) minus 12 (449312119899 minus 22226)sdot (42119899) ((radic22 ) log(radic22 )) ] + (42119899 minus 2)sdot ((radic22 ) log(radic22 )) ] + (162119899) ((23) log(23)) ]+ (202119899 minus 9)((radic156 ) log(radic156 )) ]
Table 2 Edge partition of propyl ether imine dendrimers based ondegree of end vertices of each edge
(119889119906119889V) (12) (22) (23)Number of edges 2119899+1 2119899+4 62119899 minus 6
= ln (31550257562119899 minus 1365380844)minus 1ln (31550257562119899 minus 1365380844) (minus10464703192119899+ 4497157986)
(29)
Concluding Remarks QSARs represent predictive mod-els got from utilization of statistical instruments corre-lating biological activity (including desirable therapeuticeffect and undesirable side effects) of chemicals (toxi-cantsdrugsenvironmental pollutants) with descriptors illus-trative of molecular structure as well as properties QSARsare being connected inmany disciplines for instance toxicityprediction risk assessment and regulatory decisions leadoptimization and drug discoveryThe atom-bond connectiv-ity index denoted by ABC is a molecular structure descriptorthat has remarkable application in rationalizing the stability
Discrete Dynamics in Nature and Society 9
Table 3 Edge partition of Zinc-Porphyrin dendrimers based on degree of end vertices of each edge
(119889119906119889V) (22) (23) (33) (3 4)Number of edges 162119899-4 402119899 minus 16 82119899 minus 16 4
Table 4 Edge partition of Poly(EThyleneAmidoAmine dendrimers based on degree of end vertices of each edge
(119889119906119889V) (12) (13) (22) (2 3)Number of edges 42119899 42119899 minus 2 162119899 202119899 minus 9of linear and branched alkanes and in the strain energy ofcycloalkanes [25] Weighted entropy is a generalization ofShannonrsquos entropy and is themeasure of information suppliedby a probablistic experiment whose elementary events arecharacterized both by their objective probabilities and bysome qualitative (objective or subjective) weights [29] Itis useful to rank chemicals in quantitative high-throughputscreening experiments [30] and may be used to balancethe amount of information and the degree of homogeneityassociated to a partition of data in classes [31] Weightedentropy also found applications in the coding theory [32] Formore insights about applications of entropy please see [33]In this paper we have studied weighted entropy with atomicbond connectivity edge weights which was an open problemof [34] Our next aim is to work on entropy of weightedgraphs with geometric arithmetic and sum connectivity edgeweights
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
Authors do not have any competing interests
Authorsrsquo Contributions
All authors contributed equally to this paper
Acknowledgments
This work was supported by the Dong-A University researchfund
References
[1] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 2016
[2] W Gao M R Farahani and L Shi ldquoThe forgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterranea vol32 no 1 pp 579ndash585 2016
[3] S Kang Z Iqbal M Ishaq R Sarfraz A Aslam andW NazeerldquoOn eccentricity-based topological indices and polynomials ofphosphorus-containing dendrimersrdquo Symmetry vol 10 no 7 p237 2018
[4] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe degree-based topological indices of dendrimersrdquoChemistryvol 16 no 1 pp 681ndash688 2018
[5] S M Kang W Nazeer M A Zahid A R Nizami A Aslamand M Munir ldquoM-polynomials and topological indices of hex-derived networksrdquo Physics vol 16 no 1 pp 394ndash403 2018
[6] HWiener ldquoStructural determination of paraffinboiling pointsrdquoJournal of the American Chemical Society vol 69 no 1 pp 17ndash20 1947
[7] M Randic ldquoOn characterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23 pp6609ndash6615 1975
[8] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity indexmodeling the enthalpy of formation ofalkanesrdquo Indian Journal of Chemistry vol 37 pp 849ndash855 1998
[9] M Ghorbani and M A Hosseinzadeh ldquoHosseinzadeh Com-puting index of nanostar dendrimers Opto-electronrdquoOptoelec-tronics and Advanced Materials Rapid Communications vol 4no 9 pp 1419ndash1422 2010
[10] W Nazeer A Farooq M Younas M Munir and S Kang ldquoOnMolecular Descriptors of Carbon Nanoconesrdquo Biomoleculesvol 8 no 3 p 92 2018
[11] W Gao M Younas A Farooq A Mahboob and W NazeerldquoM-Polynomials and Degree-Based Topological Indices of theCrystallographic Structure of Moleculesrdquo Biomolecules vol 8no 4 p 107 2018
[12] Y C Kwun M Munir W Nazeer S Rafique and S M KangldquoComputational Analysis of topological indices of two BoronNanotubesrdquo Scientific reports vol 8 no 1 p 14843 2018
[13] W Gao M Younas A Farooq A Virk and W Nazeer ldquoSomeReverse Degree-Based Topological Indices and Polynomials ofDendrimersrdquoMathematics vol 6 no 10 p 214 2018
[14] M Munir W Nazeer S Rafique A R Nizami and S M KangldquoSome computational aspects of boron triangular nanotubesrdquoSymmetry vol 9 no 1 2017
[15] Y C Kwun M Munir W Nazeer S Rafique and S MKang ldquoM-Polynomials and topological indices of V-PhenylenicNanotubes and Nanotorirdquo Scientific Reports vol 7 no 1 2017
[16] C E Shannon The Mathematical Theory of CommunicationThe University of Illinois Press Urbana Ill USA 1949
[17] A Mowshowitz ldquoEntropy and the complexity of graphs II Theinformation content of digraphs and infinite graphsrdquo Bulletin ofMathematical Biology vol 30 pp 225ndash240 1968
[18] A Mowshowitz ldquoEntropy and the complexity of graphs IAn index of the relative complexity of a graphrdquo Bulletin ofMathematical Biology vol 30 pp 175ndash204 1968
[19] J Korner ldquoCoding of an information source having ambiguousalphabet and the entropy of graphsrdquo the 6th Pargue Conferenceon Information Theory Statistical Decision Functions RandomProcesses Pargue Czech Republic pp 411ndash425 1973
10 Discrete Dynamics in Nature and Society
[20] M M Dehmer N N Barbarini K K Varmuza and A AGraber ldquoNovel topological descriptors for analyzing biologicalnetworksrdquo BMC Structural Biology vol 10 article no 18 2010
[21] M Dehmer and A Mowshowitz ldquoA history of graph entropymeasuresrdquo Information Sciences vol 181 no 1 pp 57ndash78 2011
[22] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropybounds for dendrimersrdquo Applied Mathematics and Computa-tion vol 242 pp 462ndash472 2014
[23] S Ji X Li and B Huo ldquoOn reformulated Zagreb indices withrespect to acyclic unicyclic and bicyclic graphsrdquo The Matchvol 72 no 3 pp 723ndash732 2014
[24] K Xu K C Das and S Balachandran ldquoMaximizing theZagreb Indices of (nm)-Graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 72 pp 641ndash6542014
[25] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquo Filomat vol 26 no 4 pp 733ndash738 2012
[26] Y Bashir A AslamM Kamran et al ldquoOn forgotten topologicalindices of some dendrimers structurerdquoMolecules vol 22 no 62017
[27] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe Degree-based Topological Indices of Dendrimersrdquo OpenChemistry 2018
[28] A Aslam Y Bashir M Rafiq F Haider N MuhammadandN Bibi ldquoThree NewOld Vertex-Degree-Based TopologicalIndices of Some Dendrimers Structurerdquo Journal of Biology vol13 no 1 2017
[29] S Guiasu ldquoWeighted entropyrdquoReports onMathematical Physicsvol 2 no 3 pp 165ndash179 1971
[30] K R Shockley ldquoUsing weighted entropy to rank chemicals inquantitative high-throughput screening experimentsrdquo Journalof Biomolecular Screening vol 19 no 3 pp 344ndash353 2014
[31] S Guiasu ldquoGrouping data by using the weighted entropyrdquoJournal of Statistical Planning and Inference vol 15 no 1 pp63ndash69 1986
[32] A Clim ldquoWeighted entropy with applicationrdquo Analele Univer-sitatii Bucuresti Matematica pp 223ndash231 2008
[33] M Ghorbani M Dehmer and S Zangi ldquoGraph operationsbased on using distance-based graph entropiesrdquo Applied Math-ematics and Computation vol 333 pp 547ndash555 2018
[34] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropy ofweighted graphs with Randic weightsrdquo Entropy vol 17 no 6 pp3710ndash3723 2015
Hindawiwwwhindawicom Volume 2018
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Applied MathematicsJournal of
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Submit your manuscripts atwwwhindawicom
4 Discrete Dynamics in Nature and Society
= log (119860119861119862) minus 12 (119860119861119862) sumradic119889119906 + 119889V minus 2119889119906119889Vsdot [log(radic119889119906 + 119889V minus 2) minus log (119889119906119889V) )]le log (119860119861119862) minus 12 [log (4 + 4 minus 2) minus log (2)]le log (119860119861119862) minus logradic3
(17)
Similarly
119868 (119866 119860119861119862) ge log (119860119861119862) minus logradic 116 (18)
Therefore
log (119860119861119862) minus logradic 116 le 119868 (119866 119860119861119862)le log (119860119861119862) minus logradic3 (19)
Corollary 7 Let119866 = (119881 119864 119908) be any complete graph of order119899 Then we have
119868 (119866 119860119861119862) le 119899radic119899 minus 22 minus log(radic 119899119899 minus 1) (20)
Proof For any complete graph G of order 119899 we have [14]119860119861119862(119866) le 119899radic(119899 minus 2)2 therefore the result119868 (119866 119860119861119862) le 119899radic119899 minus 22 minus log(radic 119899119899 minus 1) (21)
Corollary 8 Let119866 = (119881 119864 119908) be any tree of order 119899 Then wehave
119868 (119866 119860119861119862) le radic(119899 minus 1) (119899 minus 2) minus log(radic 119899119899 minus 1) (22)
Corollary 9 For a unicyclic graph
119868 (119866 119860119861119862) le radic 119899 (21198992 minus 7119899 minus 19)2 (119899 minus 1) minus log(radic 119899119899 minus 1) (23)
Corollary 10 For a Star graph
119868 (119866 119860119861119862) le radic 119899 (21198992 minus 7119899 minus 19)2 (119899 minus 1) minus log(radic 119899119899 minus 1) (24)
3 Numerical Examples
Dendrimers are man-made nanoscale compounds withunique properties that make them useful to the health and
pharmaceutical industry as both enhancements to existingproducts and as entirely new products Dendrimers areconstructed by the successive addition of layers of branch-ing groups The final generation incorporates the surfacemolecules that give the dendrimers the desired function forpharmaceutical life science chemical electronic and mate-rials applicationsDendrimers fall under the broad heading ofnanotechnology which covers the manipulation of matter inthe size range of 1-100 nanometers (one million nanometersequal one millimeter) to create compounds structures anddevices with a novel predetermined properties
In this section we discuss entropies of four famil-iar classes of dendrimers namely Porphyrin (Figure 1)Propyl ether imine (Figure 2) Zinc-Porphyrin (Figure 3)and Poly(EThyleneAmidoAmine) (Figure 4) Dendrimers Itis important to remark that all dendrimers differ in coresThese dendrimers have been studied extensively in [25ndash28]
Example 1 Let G be the Porphyrin dendrimers then usingthe edge partition given of Porphyrin dendrimers given inTable 1 we get119860119861119862 (119866) = 7726044062119899 minus 7778174591 (25)
Therefore
119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (772604119899 minus 77781)minus 12 (772604119899 minus 77781) 100381610038161003816100381611986411003816100381610038161003816 (radic63 ) log(radic63 )+ 100381610038161003816100381611986421003816100381610038161003816 (radic32 ) log(radic32 ) + 100381610038161003816100381611986431003816100381610038161003816 (radic22 ) log(radic22 )+ 100381610038161003816100381611986441003816100381610038161003816 (radic22 ) log(radic22 ) + 100381610038161003816100381611986451003816100381610038161003816 (23) log(23)+ (100381610038161003816100381611986461003816100381610038161003816 (radic156 ) log(radic156 ))= log (772604119899 minus 77781)minus 12 (772604119899 minus 77781) (2119899) (radic63 ) log(radic63 )+ (24119899) (radic32 ) log(radic32 ) + (10119899 minus 5) (radic22 ) log(radic22 ) + (48119899 minus 6) (radic22 ) log(radic22 ) + (13119899)sdot (23) log(23) + ((8119899) (radic156 ) log(radic156 ))= log (772604119899 minus 77781)minus 12 (772604119899 minus 77781) (minus23308119899 + 2695)
(26)
Discrete Dynamics in Nature and Society 5
Table 1 Edge partition of Porphyrin dendrimers based on degree of end vertices of each edge
(119889119906119889V) (13) (14) (22) (23) (33) (34)Number of edges 2n 24n 10n-5 48n-6 13n 8n
Figure 1 Porphyrin dendrimer
Example 2 Let G be the propyl ether imine dendrimerThenusing the edge partition of G given in Table 2 we get
ABC (G) = 07072119899+1 + 07072119899+4 minus 16970 + 42422119899119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906 log(radic119889119906 + 119889119906 minus 2119889119906119889119906 )
6 Discrete Dynamics in Nature and Society
O O O
OO
O
O
O
O
O
O
O
O
OO
O
O
O
O
OO
O
OO
O
O
O
O
O
O
O
O
OO
O
O O O
O
O
O
OO
O
OO
O
O
O
O
O
O
OO
O O
O
O
O
N
N
N
N
NN
N
N
N
NN
N
NNN
N
N
N N
NNN
N
N
NN
N
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2（2
N
O
N
O
N
Figure 2 Propyl ether imine dendrimer
= log (07072119899+1 + 07072119899+4 minus 16970 + 42422119899)minus 12 (07072119899+1 + 07072119899+4 minus 16970 + 42422119899) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))] + [100381610038161003816100381611986421003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986431003816100381610038161003816 ((radic22 ) log(radic22 ))] = log (07072119899+1 + 07072119899+4 minus 16970 + 42422119899)minus 12 (07072119899+1 + 07072119899+4 minus 16970 + 42422119899) [(2119899+1)((radic22 ) log(radic22 ))] + (2119899+4)((radic22 ) log(radic22 )) ]
+ (62119899 minus 6)((radic22 ) log(radic22 )) ]
= minus 1ln (070710678102119899+1 + 070710678102119899+4 minus 1697056275 + 0707106781062119899) (minus024506453602119899+1minus 024506453602119899+4 + 5881548863 minus 0245064536062119899)
(27)
Example 3 For the Zinc-Porphyrin dendrimers G usingedge partition given in Table 3 we get
ABC (G) = 449312119899 minus 22226119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906
sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (449312119899 minus 22226)minus 12 (449312119899 minus 22226) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986421003816100381610038161003816 ((radic22 ) log(radic22 ))]
Discrete Dynamics in Nature and Society 7
N N N
N
N
N
N
N
NN
N
N
N
NN
N
N
NN
N
N
N
N
N
N
N
N
NN
NNN
NNZn
N
NN
N
N
N
N
N
N
N
NN
N N N
NN
N
N
N
N
N
N
NN
N
NN
N
N
Figure 3 Zinc-Porphyrin dendrimer
+ [100381610038161003816100381611986431003816100381610038161003816 ((23) log(23))]+ [100381610038161003816100381611986441003816100381610038161003816 ((radic156 ) log(radic156 ))]= log (449312119899 minus 22226) minus 12 (449312119899 minus 22226)sdot [(162119899 minus 4)((radic22 ) log(radic22 ))] + (402119899 minus 16)sdot ((radic22 ) log(radic22 )) ] + (82119899 minus 16) ((23) log(23)) ]+ (4) ((radic156 ) log(radic156 )) ] = ln (44931313072119899
minus 2222681339)minus 1ln (44931313072119899 minus 2222681339) (minus15886094592119899
minus 8096026594)(28)
Example 4 Let G be the graph of Poly(EThyleneAmido-Amine then using edge partition given in Table 4 we get
ABC (G) = 315502119899 minus 13653119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906
8 Discrete Dynamics in Nature and Society
G5
Figure 4 Poly(EThyleneAmidoAmine dendrimer
sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (315502119899 minus 13653)minus 12 (315502119899 minus 13653) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986421003816100381610038161003816 ((radic63 ) log(radic63 )]+ [100381610038161003816100381611986431003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986441003816100381610038161003816 ((radic22 ) log(radic22 ))]= log (449312119899 minus 22226) minus 12 (449312119899 minus 22226)sdot (42119899) ((radic22 ) log(radic22 )) ] + (42119899 minus 2)sdot ((radic22 ) log(radic22 )) ] + (162119899) ((23) log(23)) ]+ (202119899 minus 9)((radic156 ) log(radic156 )) ]
Table 2 Edge partition of propyl ether imine dendrimers based ondegree of end vertices of each edge
(119889119906119889V) (12) (22) (23)Number of edges 2119899+1 2119899+4 62119899 minus 6
= ln (31550257562119899 minus 1365380844)minus 1ln (31550257562119899 minus 1365380844) (minus10464703192119899+ 4497157986)
(29)
Concluding Remarks QSARs represent predictive mod-els got from utilization of statistical instruments corre-lating biological activity (including desirable therapeuticeffect and undesirable side effects) of chemicals (toxi-cantsdrugsenvironmental pollutants) with descriptors illus-trative of molecular structure as well as properties QSARsare being connected inmany disciplines for instance toxicityprediction risk assessment and regulatory decisions leadoptimization and drug discoveryThe atom-bond connectiv-ity index denoted by ABC is a molecular structure descriptorthat has remarkable application in rationalizing the stability
Discrete Dynamics in Nature and Society 9
Table 3 Edge partition of Zinc-Porphyrin dendrimers based on degree of end vertices of each edge
(119889119906119889V) (22) (23) (33) (3 4)Number of edges 162119899-4 402119899 minus 16 82119899 minus 16 4
Table 4 Edge partition of Poly(EThyleneAmidoAmine dendrimers based on degree of end vertices of each edge
(119889119906119889V) (12) (13) (22) (2 3)Number of edges 42119899 42119899 minus 2 162119899 202119899 minus 9of linear and branched alkanes and in the strain energy ofcycloalkanes [25] Weighted entropy is a generalization ofShannonrsquos entropy and is themeasure of information suppliedby a probablistic experiment whose elementary events arecharacterized both by their objective probabilities and bysome qualitative (objective or subjective) weights [29] Itis useful to rank chemicals in quantitative high-throughputscreening experiments [30] and may be used to balancethe amount of information and the degree of homogeneityassociated to a partition of data in classes [31] Weightedentropy also found applications in the coding theory [32] Formore insights about applications of entropy please see [33]In this paper we have studied weighted entropy with atomicbond connectivity edge weights which was an open problemof [34] Our next aim is to work on entropy of weightedgraphs with geometric arithmetic and sum connectivity edgeweights
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
Authors do not have any competing interests
Authorsrsquo Contributions
All authors contributed equally to this paper
Acknowledgments
This work was supported by the Dong-A University researchfund
References
[1] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 2016
[2] W Gao M R Farahani and L Shi ldquoThe forgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterranea vol32 no 1 pp 579ndash585 2016
[3] S Kang Z Iqbal M Ishaq R Sarfraz A Aslam andW NazeerldquoOn eccentricity-based topological indices and polynomials ofphosphorus-containing dendrimersrdquo Symmetry vol 10 no 7 p237 2018
[4] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe degree-based topological indices of dendrimersrdquoChemistryvol 16 no 1 pp 681ndash688 2018
[5] S M Kang W Nazeer M A Zahid A R Nizami A Aslamand M Munir ldquoM-polynomials and topological indices of hex-derived networksrdquo Physics vol 16 no 1 pp 394ndash403 2018
[6] HWiener ldquoStructural determination of paraffinboiling pointsrdquoJournal of the American Chemical Society vol 69 no 1 pp 17ndash20 1947
[7] M Randic ldquoOn characterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23 pp6609ndash6615 1975
[8] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity indexmodeling the enthalpy of formation ofalkanesrdquo Indian Journal of Chemistry vol 37 pp 849ndash855 1998
[9] M Ghorbani and M A Hosseinzadeh ldquoHosseinzadeh Com-puting index of nanostar dendrimers Opto-electronrdquoOptoelec-tronics and Advanced Materials Rapid Communications vol 4no 9 pp 1419ndash1422 2010
[10] W Nazeer A Farooq M Younas M Munir and S Kang ldquoOnMolecular Descriptors of Carbon Nanoconesrdquo Biomoleculesvol 8 no 3 p 92 2018
[11] W Gao M Younas A Farooq A Mahboob and W NazeerldquoM-Polynomials and Degree-Based Topological Indices of theCrystallographic Structure of Moleculesrdquo Biomolecules vol 8no 4 p 107 2018
[12] Y C Kwun M Munir W Nazeer S Rafique and S M KangldquoComputational Analysis of topological indices of two BoronNanotubesrdquo Scientific reports vol 8 no 1 p 14843 2018
[13] W Gao M Younas A Farooq A Virk and W Nazeer ldquoSomeReverse Degree-Based Topological Indices and Polynomials ofDendrimersrdquoMathematics vol 6 no 10 p 214 2018
[14] M Munir W Nazeer S Rafique A R Nizami and S M KangldquoSome computational aspects of boron triangular nanotubesrdquoSymmetry vol 9 no 1 2017
[15] Y C Kwun M Munir W Nazeer S Rafique and S MKang ldquoM-Polynomials and topological indices of V-PhenylenicNanotubes and Nanotorirdquo Scientific Reports vol 7 no 1 2017
[16] C E Shannon The Mathematical Theory of CommunicationThe University of Illinois Press Urbana Ill USA 1949
[17] A Mowshowitz ldquoEntropy and the complexity of graphs II Theinformation content of digraphs and infinite graphsrdquo Bulletin ofMathematical Biology vol 30 pp 225ndash240 1968
[18] A Mowshowitz ldquoEntropy and the complexity of graphs IAn index of the relative complexity of a graphrdquo Bulletin ofMathematical Biology vol 30 pp 175ndash204 1968
[19] J Korner ldquoCoding of an information source having ambiguousalphabet and the entropy of graphsrdquo the 6th Pargue Conferenceon Information Theory Statistical Decision Functions RandomProcesses Pargue Czech Republic pp 411ndash425 1973
10 Discrete Dynamics in Nature and Society
[20] M M Dehmer N N Barbarini K K Varmuza and A AGraber ldquoNovel topological descriptors for analyzing biologicalnetworksrdquo BMC Structural Biology vol 10 article no 18 2010
[21] M Dehmer and A Mowshowitz ldquoA history of graph entropymeasuresrdquo Information Sciences vol 181 no 1 pp 57ndash78 2011
[22] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropybounds for dendrimersrdquo Applied Mathematics and Computa-tion vol 242 pp 462ndash472 2014
[23] S Ji X Li and B Huo ldquoOn reformulated Zagreb indices withrespect to acyclic unicyclic and bicyclic graphsrdquo The Matchvol 72 no 3 pp 723ndash732 2014
[24] K Xu K C Das and S Balachandran ldquoMaximizing theZagreb Indices of (nm)-Graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 72 pp 641ndash6542014
[25] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquo Filomat vol 26 no 4 pp 733ndash738 2012
[26] Y Bashir A AslamM Kamran et al ldquoOn forgotten topologicalindices of some dendrimers structurerdquoMolecules vol 22 no 62017
[27] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe Degree-based Topological Indices of Dendrimersrdquo OpenChemistry 2018
[28] A Aslam Y Bashir M Rafiq F Haider N MuhammadandN Bibi ldquoThree NewOld Vertex-Degree-Based TopologicalIndices of Some Dendrimers Structurerdquo Journal of Biology vol13 no 1 2017
[29] S Guiasu ldquoWeighted entropyrdquoReports onMathematical Physicsvol 2 no 3 pp 165ndash179 1971
[30] K R Shockley ldquoUsing weighted entropy to rank chemicals inquantitative high-throughput screening experimentsrdquo Journalof Biomolecular Screening vol 19 no 3 pp 344ndash353 2014
[31] S Guiasu ldquoGrouping data by using the weighted entropyrdquoJournal of Statistical Planning and Inference vol 15 no 1 pp63ndash69 1986
[32] A Clim ldquoWeighted entropy with applicationrdquo Analele Univer-sitatii Bucuresti Matematica pp 223ndash231 2008
[33] M Ghorbani M Dehmer and S Zangi ldquoGraph operationsbased on using distance-based graph entropiesrdquo Applied Math-ematics and Computation vol 333 pp 547ndash555 2018
[34] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropy ofweighted graphs with Randic weightsrdquo Entropy vol 17 no 6 pp3710ndash3723 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Discrete Dynamics in Nature and Society 5
Table 1 Edge partition of Porphyrin dendrimers based on degree of end vertices of each edge
(119889119906119889V) (13) (14) (22) (23) (33) (34)Number of edges 2n 24n 10n-5 48n-6 13n 8n
Figure 1 Porphyrin dendrimer
Example 2 Let G be the propyl ether imine dendrimerThenusing the edge partition of G given in Table 2 we get
ABC (G) = 07072119899+1 + 07072119899+4 minus 16970 + 42422119899119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906 log(radic119889119906 + 119889119906 minus 2119889119906119889119906 )
6 Discrete Dynamics in Nature and Society
O O O
OO
O
O
O
O
O
O
O
O
OO
O
O
O
O
OO
O
OO
O
O
O
O
O
O
O
O
OO
O
O O O
O
O
O
OO
O
OO
O
O
O
O
O
O
OO
O O
O
O
O
N
N
N
N
NN
N
N
N
NN
N
NNN
N
N
N N
NNN
N
N
NN
N
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2（2
N
O
N
O
N
Figure 2 Propyl ether imine dendrimer
= log (07072119899+1 + 07072119899+4 minus 16970 + 42422119899)minus 12 (07072119899+1 + 07072119899+4 minus 16970 + 42422119899) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))] + [100381610038161003816100381611986421003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986431003816100381610038161003816 ((radic22 ) log(radic22 ))] = log (07072119899+1 + 07072119899+4 minus 16970 + 42422119899)minus 12 (07072119899+1 + 07072119899+4 minus 16970 + 42422119899) [(2119899+1)((radic22 ) log(radic22 ))] + (2119899+4)((radic22 ) log(radic22 )) ]
+ (62119899 minus 6)((radic22 ) log(radic22 )) ]
= minus 1ln (070710678102119899+1 + 070710678102119899+4 minus 1697056275 + 0707106781062119899) (minus024506453602119899+1minus 024506453602119899+4 + 5881548863 minus 0245064536062119899)
(27)
Example 3 For the Zinc-Porphyrin dendrimers G usingedge partition given in Table 3 we get
ABC (G) = 449312119899 minus 22226119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906
sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (449312119899 minus 22226)minus 12 (449312119899 minus 22226) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986421003816100381610038161003816 ((radic22 ) log(radic22 ))]
Discrete Dynamics in Nature and Society 7
N N N
N
N
N
N
N
NN
N
N
N
NN
N
N
NN
N
N
N
N
N
N
N
N
NN
NNN
NNZn
N
NN
N
N
N
N
N
N
N
NN
N N N
NN
N
N
N
N
N
N
NN
N
NN
N
N
Figure 3 Zinc-Porphyrin dendrimer
+ [100381610038161003816100381611986431003816100381610038161003816 ((23) log(23))]+ [100381610038161003816100381611986441003816100381610038161003816 ((radic156 ) log(radic156 ))]= log (449312119899 minus 22226) minus 12 (449312119899 minus 22226)sdot [(162119899 minus 4)((radic22 ) log(radic22 ))] + (402119899 minus 16)sdot ((radic22 ) log(radic22 )) ] + (82119899 minus 16) ((23) log(23)) ]+ (4) ((radic156 ) log(radic156 )) ] = ln (44931313072119899
minus 2222681339)minus 1ln (44931313072119899 minus 2222681339) (minus15886094592119899
minus 8096026594)(28)
Example 4 Let G be the graph of Poly(EThyleneAmido-Amine then using edge partition given in Table 4 we get
ABC (G) = 315502119899 minus 13653119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906
8 Discrete Dynamics in Nature and Society
G5
Figure 4 Poly(EThyleneAmidoAmine dendrimer
sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (315502119899 minus 13653)minus 12 (315502119899 minus 13653) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986421003816100381610038161003816 ((radic63 ) log(radic63 )]+ [100381610038161003816100381611986431003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986441003816100381610038161003816 ((radic22 ) log(radic22 ))]= log (449312119899 minus 22226) minus 12 (449312119899 minus 22226)sdot (42119899) ((radic22 ) log(radic22 )) ] + (42119899 minus 2)sdot ((radic22 ) log(radic22 )) ] + (162119899) ((23) log(23)) ]+ (202119899 minus 9)((radic156 ) log(radic156 )) ]
Table 2 Edge partition of propyl ether imine dendrimers based ondegree of end vertices of each edge
(119889119906119889V) (12) (22) (23)Number of edges 2119899+1 2119899+4 62119899 minus 6
= ln (31550257562119899 minus 1365380844)minus 1ln (31550257562119899 minus 1365380844) (minus10464703192119899+ 4497157986)
(29)
Concluding Remarks QSARs represent predictive mod-els got from utilization of statistical instruments corre-lating biological activity (including desirable therapeuticeffect and undesirable side effects) of chemicals (toxi-cantsdrugsenvironmental pollutants) with descriptors illus-trative of molecular structure as well as properties QSARsare being connected inmany disciplines for instance toxicityprediction risk assessment and regulatory decisions leadoptimization and drug discoveryThe atom-bond connectiv-ity index denoted by ABC is a molecular structure descriptorthat has remarkable application in rationalizing the stability
Discrete Dynamics in Nature and Society 9
Table 3 Edge partition of Zinc-Porphyrin dendrimers based on degree of end vertices of each edge
(119889119906119889V) (22) (23) (33) (3 4)Number of edges 162119899-4 402119899 minus 16 82119899 minus 16 4
Table 4 Edge partition of Poly(EThyleneAmidoAmine dendrimers based on degree of end vertices of each edge
(119889119906119889V) (12) (13) (22) (2 3)Number of edges 42119899 42119899 minus 2 162119899 202119899 minus 9of linear and branched alkanes and in the strain energy ofcycloalkanes [25] Weighted entropy is a generalization ofShannonrsquos entropy and is themeasure of information suppliedby a probablistic experiment whose elementary events arecharacterized both by their objective probabilities and bysome qualitative (objective or subjective) weights [29] Itis useful to rank chemicals in quantitative high-throughputscreening experiments [30] and may be used to balancethe amount of information and the degree of homogeneityassociated to a partition of data in classes [31] Weightedentropy also found applications in the coding theory [32] Formore insights about applications of entropy please see [33]In this paper we have studied weighted entropy with atomicbond connectivity edge weights which was an open problemof [34] Our next aim is to work on entropy of weightedgraphs with geometric arithmetic and sum connectivity edgeweights
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
Authors do not have any competing interests
Authorsrsquo Contributions
All authors contributed equally to this paper
Acknowledgments
This work was supported by the Dong-A University researchfund
References
[1] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 2016
[2] W Gao M R Farahani and L Shi ldquoThe forgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterranea vol32 no 1 pp 579ndash585 2016
[3] S Kang Z Iqbal M Ishaq R Sarfraz A Aslam andW NazeerldquoOn eccentricity-based topological indices and polynomials ofphosphorus-containing dendrimersrdquo Symmetry vol 10 no 7 p237 2018
[4] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe degree-based topological indices of dendrimersrdquoChemistryvol 16 no 1 pp 681ndash688 2018
[5] S M Kang W Nazeer M A Zahid A R Nizami A Aslamand M Munir ldquoM-polynomials and topological indices of hex-derived networksrdquo Physics vol 16 no 1 pp 394ndash403 2018
[6] HWiener ldquoStructural determination of paraffinboiling pointsrdquoJournal of the American Chemical Society vol 69 no 1 pp 17ndash20 1947
[7] M Randic ldquoOn characterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23 pp6609ndash6615 1975
[8] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity indexmodeling the enthalpy of formation ofalkanesrdquo Indian Journal of Chemistry vol 37 pp 849ndash855 1998
[9] M Ghorbani and M A Hosseinzadeh ldquoHosseinzadeh Com-puting index of nanostar dendrimers Opto-electronrdquoOptoelec-tronics and Advanced Materials Rapid Communications vol 4no 9 pp 1419ndash1422 2010
[10] W Nazeer A Farooq M Younas M Munir and S Kang ldquoOnMolecular Descriptors of Carbon Nanoconesrdquo Biomoleculesvol 8 no 3 p 92 2018
[11] W Gao M Younas A Farooq A Mahboob and W NazeerldquoM-Polynomials and Degree-Based Topological Indices of theCrystallographic Structure of Moleculesrdquo Biomolecules vol 8no 4 p 107 2018
[12] Y C Kwun M Munir W Nazeer S Rafique and S M KangldquoComputational Analysis of topological indices of two BoronNanotubesrdquo Scientific reports vol 8 no 1 p 14843 2018
[13] W Gao M Younas A Farooq A Virk and W Nazeer ldquoSomeReverse Degree-Based Topological Indices and Polynomials ofDendrimersrdquoMathematics vol 6 no 10 p 214 2018
[14] M Munir W Nazeer S Rafique A R Nizami and S M KangldquoSome computational aspects of boron triangular nanotubesrdquoSymmetry vol 9 no 1 2017
[15] Y C Kwun M Munir W Nazeer S Rafique and S MKang ldquoM-Polynomials and topological indices of V-PhenylenicNanotubes and Nanotorirdquo Scientific Reports vol 7 no 1 2017
[16] C E Shannon The Mathematical Theory of CommunicationThe University of Illinois Press Urbana Ill USA 1949
[17] A Mowshowitz ldquoEntropy and the complexity of graphs II Theinformation content of digraphs and infinite graphsrdquo Bulletin ofMathematical Biology vol 30 pp 225ndash240 1968
[18] A Mowshowitz ldquoEntropy and the complexity of graphs IAn index of the relative complexity of a graphrdquo Bulletin ofMathematical Biology vol 30 pp 175ndash204 1968
[19] J Korner ldquoCoding of an information source having ambiguousalphabet and the entropy of graphsrdquo the 6th Pargue Conferenceon Information Theory Statistical Decision Functions RandomProcesses Pargue Czech Republic pp 411ndash425 1973
10 Discrete Dynamics in Nature and Society
[20] M M Dehmer N N Barbarini K K Varmuza and A AGraber ldquoNovel topological descriptors for analyzing biologicalnetworksrdquo BMC Structural Biology vol 10 article no 18 2010
[21] M Dehmer and A Mowshowitz ldquoA history of graph entropymeasuresrdquo Information Sciences vol 181 no 1 pp 57ndash78 2011
[22] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropybounds for dendrimersrdquo Applied Mathematics and Computa-tion vol 242 pp 462ndash472 2014
[23] S Ji X Li and B Huo ldquoOn reformulated Zagreb indices withrespect to acyclic unicyclic and bicyclic graphsrdquo The Matchvol 72 no 3 pp 723ndash732 2014
[24] K Xu K C Das and S Balachandran ldquoMaximizing theZagreb Indices of (nm)-Graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 72 pp 641ndash6542014
[25] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquo Filomat vol 26 no 4 pp 733ndash738 2012
[26] Y Bashir A AslamM Kamran et al ldquoOn forgotten topologicalindices of some dendrimers structurerdquoMolecules vol 22 no 62017
[27] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe Degree-based Topological Indices of Dendrimersrdquo OpenChemistry 2018
[28] A Aslam Y Bashir M Rafiq F Haider N MuhammadandN Bibi ldquoThree NewOld Vertex-Degree-Based TopologicalIndices of Some Dendrimers Structurerdquo Journal of Biology vol13 no 1 2017
[29] S Guiasu ldquoWeighted entropyrdquoReports onMathematical Physicsvol 2 no 3 pp 165ndash179 1971
[30] K R Shockley ldquoUsing weighted entropy to rank chemicals inquantitative high-throughput screening experimentsrdquo Journalof Biomolecular Screening vol 19 no 3 pp 344ndash353 2014
[31] S Guiasu ldquoGrouping data by using the weighted entropyrdquoJournal of Statistical Planning and Inference vol 15 no 1 pp63ndash69 1986
[32] A Clim ldquoWeighted entropy with applicationrdquo Analele Univer-sitatii Bucuresti Matematica pp 223ndash231 2008
[33] M Ghorbani M Dehmer and S Zangi ldquoGraph operationsbased on using distance-based graph entropiesrdquo Applied Math-ematics and Computation vol 333 pp 547ndash555 2018
[34] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropy ofweighted graphs with Randic weightsrdquo Entropy vol 17 no 6 pp3710ndash3723 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
6 Discrete Dynamics in Nature and Society
O O O
OO
O
O
O
O
O
O
O
O
OO
O
O
O
O
OO
O
OO
O
O
O
O
O
O
O
O
OO
O
O O O
O
O
O
OO
O
OO
O
O
O
O
O
O
OO
O O
O
O
O
N
N
N
N
NN
N
N
N
NN
N
NNN
N
N
N N
NNN
N
N
NN
N
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2
（2（2
N
O
N
O
N
Figure 2 Propyl ether imine dendrimer
= log (07072119899+1 + 07072119899+4 minus 16970 + 42422119899)minus 12 (07072119899+1 + 07072119899+4 minus 16970 + 42422119899) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))] + [100381610038161003816100381611986421003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986431003816100381610038161003816 ((radic22 ) log(radic22 ))] = log (07072119899+1 + 07072119899+4 minus 16970 + 42422119899)minus 12 (07072119899+1 + 07072119899+4 minus 16970 + 42422119899) [(2119899+1)((radic22 ) log(radic22 ))] + (2119899+4)((radic22 ) log(radic22 )) ]
+ (62119899 minus 6)((radic22 ) log(radic22 )) ]
= minus 1ln (070710678102119899+1 + 070710678102119899+4 minus 1697056275 + 0707106781062119899) (minus024506453602119899+1minus 024506453602119899+4 + 5881548863 minus 0245064536062119899)
(27)
Example 3 For the Zinc-Porphyrin dendrimers G usingedge partition given in Table 3 we get
ABC (G) = 449312119899 minus 22226119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906
sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (449312119899 minus 22226)minus 12 (449312119899 minus 22226) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986421003816100381610038161003816 ((radic22 ) log(radic22 ))]
Discrete Dynamics in Nature and Society 7
N N N
N
N
N
N
N
NN
N
N
N
NN
N
N
NN
N
N
N
N
N
N
N
N
NN
NNN
NNZn
N
NN
N
N
N
N
N
N
N
NN
N N N
NN
N
N
N
N
N
N
NN
N
NN
N
N
Figure 3 Zinc-Porphyrin dendrimer
+ [100381610038161003816100381611986431003816100381610038161003816 ((23) log(23))]+ [100381610038161003816100381611986441003816100381610038161003816 ((radic156 ) log(radic156 ))]= log (449312119899 minus 22226) minus 12 (449312119899 minus 22226)sdot [(162119899 minus 4)((radic22 ) log(radic22 ))] + (402119899 minus 16)sdot ((radic22 ) log(radic22 )) ] + (82119899 minus 16) ((23) log(23)) ]+ (4) ((radic156 ) log(radic156 )) ] = ln (44931313072119899
minus 2222681339)minus 1ln (44931313072119899 minus 2222681339) (minus15886094592119899
minus 8096026594)(28)
Example 4 Let G be the graph of Poly(EThyleneAmido-Amine then using edge partition given in Table 4 we get
ABC (G) = 315502119899 minus 13653119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906
8 Discrete Dynamics in Nature and Society
G5
Figure 4 Poly(EThyleneAmidoAmine dendrimer
sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (315502119899 minus 13653)minus 12 (315502119899 minus 13653) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986421003816100381610038161003816 ((radic63 ) log(radic63 )]+ [100381610038161003816100381611986431003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986441003816100381610038161003816 ((radic22 ) log(radic22 ))]= log (449312119899 minus 22226) minus 12 (449312119899 minus 22226)sdot (42119899) ((radic22 ) log(radic22 )) ] + (42119899 minus 2)sdot ((radic22 ) log(radic22 )) ] + (162119899) ((23) log(23)) ]+ (202119899 minus 9)((radic156 ) log(radic156 )) ]
Table 2 Edge partition of propyl ether imine dendrimers based ondegree of end vertices of each edge
(119889119906119889V) (12) (22) (23)Number of edges 2119899+1 2119899+4 62119899 minus 6
= ln (31550257562119899 minus 1365380844)minus 1ln (31550257562119899 minus 1365380844) (minus10464703192119899+ 4497157986)
(29)
Concluding Remarks QSARs represent predictive mod-els got from utilization of statistical instruments corre-lating biological activity (including desirable therapeuticeffect and undesirable side effects) of chemicals (toxi-cantsdrugsenvironmental pollutants) with descriptors illus-trative of molecular structure as well as properties QSARsare being connected inmany disciplines for instance toxicityprediction risk assessment and regulatory decisions leadoptimization and drug discoveryThe atom-bond connectiv-ity index denoted by ABC is a molecular structure descriptorthat has remarkable application in rationalizing the stability
Discrete Dynamics in Nature and Society 9
Table 3 Edge partition of Zinc-Porphyrin dendrimers based on degree of end vertices of each edge
(119889119906119889V) (22) (23) (33) (3 4)Number of edges 162119899-4 402119899 minus 16 82119899 minus 16 4
Table 4 Edge partition of Poly(EThyleneAmidoAmine dendrimers based on degree of end vertices of each edge
(119889119906119889V) (12) (13) (22) (2 3)Number of edges 42119899 42119899 minus 2 162119899 202119899 minus 9of linear and branched alkanes and in the strain energy ofcycloalkanes [25] Weighted entropy is a generalization ofShannonrsquos entropy and is themeasure of information suppliedby a probablistic experiment whose elementary events arecharacterized both by their objective probabilities and bysome qualitative (objective or subjective) weights [29] Itis useful to rank chemicals in quantitative high-throughputscreening experiments [30] and may be used to balancethe amount of information and the degree of homogeneityassociated to a partition of data in classes [31] Weightedentropy also found applications in the coding theory [32] Formore insights about applications of entropy please see [33]In this paper we have studied weighted entropy with atomicbond connectivity edge weights which was an open problemof [34] Our next aim is to work on entropy of weightedgraphs with geometric arithmetic and sum connectivity edgeweights
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
Authors do not have any competing interests
Authorsrsquo Contributions
All authors contributed equally to this paper
Acknowledgments
This work was supported by the Dong-A University researchfund
References
[1] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 2016
[2] W Gao M R Farahani and L Shi ldquoThe forgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterranea vol32 no 1 pp 579ndash585 2016
[3] S Kang Z Iqbal M Ishaq R Sarfraz A Aslam andW NazeerldquoOn eccentricity-based topological indices and polynomials ofphosphorus-containing dendrimersrdquo Symmetry vol 10 no 7 p237 2018
[4] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe degree-based topological indices of dendrimersrdquoChemistryvol 16 no 1 pp 681ndash688 2018
[5] S M Kang W Nazeer M A Zahid A R Nizami A Aslamand M Munir ldquoM-polynomials and topological indices of hex-derived networksrdquo Physics vol 16 no 1 pp 394ndash403 2018
[6] HWiener ldquoStructural determination of paraffinboiling pointsrdquoJournal of the American Chemical Society vol 69 no 1 pp 17ndash20 1947
[7] M Randic ldquoOn characterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23 pp6609ndash6615 1975
[8] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity indexmodeling the enthalpy of formation ofalkanesrdquo Indian Journal of Chemistry vol 37 pp 849ndash855 1998
[9] M Ghorbani and M A Hosseinzadeh ldquoHosseinzadeh Com-puting index of nanostar dendrimers Opto-electronrdquoOptoelec-tronics and Advanced Materials Rapid Communications vol 4no 9 pp 1419ndash1422 2010
[10] W Nazeer A Farooq M Younas M Munir and S Kang ldquoOnMolecular Descriptors of Carbon Nanoconesrdquo Biomoleculesvol 8 no 3 p 92 2018
[11] W Gao M Younas A Farooq A Mahboob and W NazeerldquoM-Polynomials and Degree-Based Topological Indices of theCrystallographic Structure of Moleculesrdquo Biomolecules vol 8no 4 p 107 2018
[12] Y C Kwun M Munir W Nazeer S Rafique and S M KangldquoComputational Analysis of topological indices of two BoronNanotubesrdquo Scientific reports vol 8 no 1 p 14843 2018
[13] W Gao M Younas A Farooq A Virk and W Nazeer ldquoSomeReverse Degree-Based Topological Indices and Polynomials ofDendrimersrdquoMathematics vol 6 no 10 p 214 2018
[14] M Munir W Nazeer S Rafique A R Nizami and S M KangldquoSome computational aspects of boron triangular nanotubesrdquoSymmetry vol 9 no 1 2017
[15] Y C Kwun M Munir W Nazeer S Rafique and S MKang ldquoM-Polynomials and topological indices of V-PhenylenicNanotubes and Nanotorirdquo Scientific Reports vol 7 no 1 2017
[16] C E Shannon The Mathematical Theory of CommunicationThe University of Illinois Press Urbana Ill USA 1949
[17] A Mowshowitz ldquoEntropy and the complexity of graphs II Theinformation content of digraphs and infinite graphsrdquo Bulletin ofMathematical Biology vol 30 pp 225ndash240 1968
[18] A Mowshowitz ldquoEntropy and the complexity of graphs IAn index of the relative complexity of a graphrdquo Bulletin ofMathematical Biology vol 30 pp 175ndash204 1968
[19] J Korner ldquoCoding of an information source having ambiguousalphabet and the entropy of graphsrdquo the 6th Pargue Conferenceon Information Theory Statistical Decision Functions RandomProcesses Pargue Czech Republic pp 411ndash425 1973
10 Discrete Dynamics in Nature and Society
[20] M M Dehmer N N Barbarini K K Varmuza and A AGraber ldquoNovel topological descriptors for analyzing biologicalnetworksrdquo BMC Structural Biology vol 10 article no 18 2010
[21] M Dehmer and A Mowshowitz ldquoA history of graph entropymeasuresrdquo Information Sciences vol 181 no 1 pp 57ndash78 2011
[22] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropybounds for dendrimersrdquo Applied Mathematics and Computa-tion vol 242 pp 462ndash472 2014
[23] S Ji X Li and B Huo ldquoOn reformulated Zagreb indices withrespect to acyclic unicyclic and bicyclic graphsrdquo The Matchvol 72 no 3 pp 723ndash732 2014
[24] K Xu K C Das and S Balachandran ldquoMaximizing theZagreb Indices of (nm)-Graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 72 pp 641ndash6542014
[25] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquo Filomat vol 26 no 4 pp 733ndash738 2012
[26] Y Bashir A AslamM Kamran et al ldquoOn forgotten topologicalindices of some dendrimers structurerdquoMolecules vol 22 no 62017
[27] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe Degree-based Topological Indices of Dendrimersrdquo OpenChemistry 2018
[28] A Aslam Y Bashir M Rafiq F Haider N MuhammadandN Bibi ldquoThree NewOld Vertex-Degree-Based TopologicalIndices of Some Dendrimers Structurerdquo Journal of Biology vol13 no 1 2017
[29] S Guiasu ldquoWeighted entropyrdquoReports onMathematical Physicsvol 2 no 3 pp 165ndash179 1971
[30] K R Shockley ldquoUsing weighted entropy to rank chemicals inquantitative high-throughput screening experimentsrdquo Journalof Biomolecular Screening vol 19 no 3 pp 344ndash353 2014
[31] S Guiasu ldquoGrouping data by using the weighted entropyrdquoJournal of Statistical Planning and Inference vol 15 no 1 pp63ndash69 1986
[32] A Clim ldquoWeighted entropy with applicationrdquo Analele Univer-sitatii Bucuresti Matematica pp 223ndash231 2008
[33] M Ghorbani M Dehmer and S Zangi ldquoGraph operationsbased on using distance-based graph entropiesrdquo Applied Math-ematics and Computation vol 333 pp 547ndash555 2018
[34] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropy ofweighted graphs with Randic weightsrdquo Entropy vol 17 no 6 pp3710ndash3723 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Discrete Dynamics in Nature and Society 7
N N N
N
N
N
N
N
NN
N
N
N
NN
N
N
NN
N
N
N
N
N
N
N
N
NN
NNN
NNZn
N
NN
N
N
N
N
N
N
N
NN
N N N
NN
N
N
N
N
N
N
NN
N
NN
N
N
Figure 3 Zinc-Porphyrin dendrimer
+ [100381610038161003816100381611986431003816100381610038161003816 ((23) log(23))]+ [100381610038161003816100381611986441003816100381610038161003816 ((radic156 ) log(radic156 ))]= log (449312119899 minus 22226) minus 12 (449312119899 minus 22226)sdot [(162119899 minus 4)((radic22 ) log(radic22 ))] + (402119899 minus 16)sdot ((radic22 ) log(radic22 )) ] + (82119899 minus 16) ((23) log(23)) ]+ (4) ((radic156 ) log(radic156 )) ] = ln (44931313072119899
minus 2222681339)minus 1ln (44931313072119899 minus 2222681339) (minus15886094592119899
minus 8096026594)(28)
Example 4 Let G be the graph of Poly(EThyleneAmido-Amine then using edge partition given in Table 4 we get
ABC (G) = 315502119899 minus 13653119868 (119866 119860119861119862) = log (119860119861119862) minus 12119860119861119862 sum (radic119889119906 + 119889119906 minus 2119889119906119889119906
8 Discrete Dynamics in Nature and Society
G5
Figure 4 Poly(EThyleneAmidoAmine dendrimer
sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (315502119899 minus 13653)minus 12 (315502119899 minus 13653) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986421003816100381610038161003816 ((radic63 ) log(radic63 )]+ [100381610038161003816100381611986431003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986441003816100381610038161003816 ((radic22 ) log(radic22 ))]= log (449312119899 minus 22226) minus 12 (449312119899 minus 22226)sdot (42119899) ((radic22 ) log(radic22 )) ] + (42119899 minus 2)sdot ((radic22 ) log(radic22 )) ] + (162119899) ((23) log(23)) ]+ (202119899 minus 9)((radic156 ) log(radic156 )) ]
Table 2 Edge partition of propyl ether imine dendrimers based ondegree of end vertices of each edge
(119889119906119889V) (12) (22) (23)Number of edges 2119899+1 2119899+4 62119899 minus 6
= ln (31550257562119899 minus 1365380844)minus 1ln (31550257562119899 minus 1365380844) (minus10464703192119899+ 4497157986)
(29)
Concluding Remarks QSARs represent predictive mod-els got from utilization of statistical instruments corre-lating biological activity (including desirable therapeuticeffect and undesirable side effects) of chemicals (toxi-cantsdrugsenvironmental pollutants) with descriptors illus-trative of molecular structure as well as properties QSARsare being connected inmany disciplines for instance toxicityprediction risk assessment and regulatory decisions leadoptimization and drug discoveryThe atom-bond connectiv-ity index denoted by ABC is a molecular structure descriptorthat has remarkable application in rationalizing the stability
Discrete Dynamics in Nature and Society 9
Table 3 Edge partition of Zinc-Porphyrin dendrimers based on degree of end vertices of each edge
(119889119906119889V) (22) (23) (33) (3 4)Number of edges 162119899-4 402119899 minus 16 82119899 minus 16 4
Table 4 Edge partition of Poly(EThyleneAmidoAmine dendrimers based on degree of end vertices of each edge
(119889119906119889V) (12) (13) (22) (2 3)Number of edges 42119899 42119899 minus 2 162119899 202119899 minus 9of linear and branched alkanes and in the strain energy ofcycloalkanes [25] Weighted entropy is a generalization ofShannonrsquos entropy and is themeasure of information suppliedby a probablistic experiment whose elementary events arecharacterized both by their objective probabilities and bysome qualitative (objective or subjective) weights [29] Itis useful to rank chemicals in quantitative high-throughputscreening experiments [30] and may be used to balancethe amount of information and the degree of homogeneityassociated to a partition of data in classes [31] Weightedentropy also found applications in the coding theory [32] Formore insights about applications of entropy please see [33]In this paper we have studied weighted entropy with atomicbond connectivity edge weights which was an open problemof [34] Our next aim is to work on entropy of weightedgraphs with geometric arithmetic and sum connectivity edgeweights
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
Authors do not have any competing interests
Authorsrsquo Contributions
All authors contributed equally to this paper
Acknowledgments
This work was supported by the Dong-A University researchfund
References
[1] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 2016
[2] W Gao M R Farahani and L Shi ldquoThe forgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterranea vol32 no 1 pp 579ndash585 2016
[3] S Kang Z Iqbal M Ishaq R Sarfraz A Aslam andW NazeerldquoOn eccentricity-based topological indices and polynomials ofphosphorus-containing dendrimersrdquo Symmetry vol 10 no 7 p237 2018
[4] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe degree-based topological indices of dendrimersrdquoChemistryvol 16 no 1 pp 681ndash688 2018
[5] S M Kang W Nazeer M A Zahid A R Nizami A Aslamand M Munir ldquoM-polynomials and topological indices of hex-derived networksrdquo Physics vol 16 no 1 pp 394ndash403 2018
[6] HWiener ldquoStructural determination of paraffinboiling pointsrdquoJournal of the American Chemical Society vol 69 no 1 pp 17ndash20 1947
[7] M Randic ldquoOn characterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23 pp6609ndash6615 1975
[8] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity indexmodeling the enthalpy of formation ofalkanesrdquo Indian Journal of Chemistry vol 37 pp 849ndash855 1998
[9] M Ghorbani and M A Hosseinzadeh ldquoHosseinzadeh Com-puting index of nanostar dendrimers Opto-electronrdquoOptoelec-tronics and Advanced Materials Rapid Communications vol 4no 9 pp 1419ndash1422 2010
[10] W Nazeer A Farooq M Younas M Munir and S Kang ldquoOnMolecular Descriptors of Carbon Nanoconesrdquo Biomoleculesvol 8 no 3 p 92 2018
[11] W Gao M Younas A Farooq A Mahboob and W NazeerldquoM-Polynomials and Degree-Based Topological Indices of theCrystallographic Structure of Moleculesrdquo Biomolecules vol 8no 4 p 107 2018
[12] Y C Kwun M Munir W Nazeer S Rafique and S M KangldquoComputational Analysis of topological indices of two BoronNanotubesrdquo Scientific reports vol 8 no 1 p 14843 2018
[13] W Gao M Younas A Farooq A Virk and W Nazeer ldquoSomeReverse Degree-Based Topological Indices and Polynomials ofDendrimersrdquoMathematics vol 6 no 10 p 214 2018
[14] M Munir W Nazeer S Rafique A R Nizami and S M KangldquoSome computational aspects of boron triangular nanotubesrdquoSymmetry vol 9 no 1 2017
[15] Y C Kwun M Munir W Nazeer S Rafique and S MKang ldquoM-Polynomials and topological indices of V-PhenylenicNanotubes and Nanotorirdquo Scientific Reports vol 7 no 1 2017
[16] C E Shannon The Mathematical Theory of CommunicationThe University of Illinois Press Urbana Ill USA 1949
[17] A Mowshowitz ldquoEntropy and the complexity of graphs II Theinformation content of digraphs and infinite graphsrdquo Bulletin ofMathematical Biology vol 30 pp 225ndash240 1968
[18] A Mowshowitz ldquoEntropy and the complexity of graphs IAn index of the relative complexity of a graphrdquo Bulletin ofMathematical Biology vol 30 pp 175ndash204 1968
[19] J Korner ldquoCoding of an information source having ambiguousalphabet and the entropy of graphsrdquo the 6th Pargue Conferenceon Information Theory Statistical Decision Functions RandomProcesses Pargue Czech Republic pp 411ndash425 1973
10 Discrete Dynamics in Nature and Society
[20] M M Dehmer N N Barbarini K K Varmuza and A AGraber ldquoNovel topological descriptors for analyzing biologicalnetworksrdquo BMC Structural Biology vol 10 article no 18 2010
[21] M Dehmer and A Mowshowitz ldquoA history of graph entropymeasuresrdquo Information Sciences vol 181 no 1 pp 57ndash78 2011
[22] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropybounds for dendrimersrdquo Applied Mathematics and Computa-tion vol 242 pp 462ndash472 2014
[23] S Ji X Li and B Huo ldquoOn reformulated Zagreb indices withrespect to acyclic unicyclic and bicyclic graphsrdquo The Matchvol 72 no 3 pp 723ndash732 2014
[24] K Xu K C Das and S Balachandran ldquoMaximizing theZagreb Indices of (nm)-Graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 72 pp 641ndash6542014
[25] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquo Filomat vol 26 no 4 pp 733ndash738 2012
[26] Y Bashir A AslamM Kamran et al ldquoOn forgotten topologicalindices of some dendrimers structurerdquoMolecules vol 22 no 62017
[27] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe Degree-based Topological Indices of Dendrimersrdquo OpenChemistry 2018
[28] A Aslam Y Bashir M Rafiq F Haider N MuhammadandN Bibi ldquoThree NewOld Vertex-Degree-Based TopologicalIndices of Some Dendrimers Structurerdquo Journal of Biology vol13 no 1 2017
[29] S Guiasu ldquoWeighted entropyrdquoReports onMathematical Physicsvol 2 no 3 pp 165ndash179 1971
[30] K R Shockley ldquoUsing weighted entropy to rank chemicals inquantitative high-throughput screening experimentsrdquo Journalof Biomolecular Screening vol 19 no 3 pp 344ndash353 2014
[31] S Guiasu ldquoGrouping data by using the weighted entropyrdquoJournal of Statistical Planning and Inference vol 15 no 1 pp63ndash69 1986
[32] A Clim ldquoWeighted entropy with applicationrdquo Analele Univer-sitatii Bucuresti Matematica pp 223ndash231 2008
[33] M Ghorbani M Dehmer and S Zangi ldquoGraph operationsbased on using distance-based graph entropiesrdquo Applied Math-ematics and Computation vol 333 pp 547ndash555 2018
[34] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropy ofweighted graphs with Randic weightsrdquo Entropy vol 17 no 6 pp3710ndash3723 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
8 Discrete Dynamics in Nature and Society
G5
Figure 4 Poly(EThyleneAmidoAmine dendrimer
sdot log(radic119889119906 + 119889119906 minus 2119889119906119889119906 ) = log (315502119899 minus 13653)minus 12 (315502119899 minus 13653) [100381610038161003816100381611986411003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986421003816100381610038161003816 ((radic63 ) log(radic63 )]+ [100381610038161003816100381611986431003816100381610038161003816 ((radic22 ) log(radic22 ))]+ [100381610038161003816100381611986441003816100381610038161003816 ((radic22 ) log(radic22 ))]= log (449312119899 minus 22226) minus 12 (449312119899 minus 22226)sdot (42119899) ((radic22 ) log(radic22 )) ] + (42119899 minus 2)sdot ((radic22 ) log(radic22 )) ] + (162119899) ((23) log(23)) ]+ (202119899 minus 9)((radic156 ) log(radic156 )) ]
Table 2 Edge partition of propyl ether imine dendrimers based ondegree of end vertices of each edge
(119889119906119889V) (12) (22) (23)Number of edges 2119899+1 2119899+4 62119899 minus 6
= ln (31550257562119899 minus 1365380844)minus 1ln (31550257562119899 minus 1365380844) (minus10464703192119899+ 4497157986)
(29)
Concluding Remarks QSARs represent predictive mod-els got from utilization of statistical instruments corre-lating biological activity (including desirable therapeuticeffect and undesirable side effects) of chemicals (toxi-cantsdrugsenvironmental pollutants) with descriptors illus-trative of molecular structure as well as properties QSARsare being connected inmany disciplines for instance toxicityprediction risk assessment and regulatory decisions leadoptimization and drug discoveryThe atom-bond connectiv-ity index denoted by ABC is a molecular structure descriptorthat has remarkable application in rationalizing the stability
Discrete Dynamics in Nature and Society 9
Table 3 Edge partition of Zinc-Porphyrin dendrimers based on degree of end vertices of each edge
(119889119906119889V) (22) (23) (33) (3 4)Number of edges 162119899-4 402119899 minus 16 82119899 minus 16 4
Table 4 Edge partition of Poly(EThyleneAmidoAmine dendrimers based on degree of end vertices of each edge
(119889119906119889V) (12) (13) (22) (2 3)Number of edges 42119899 42119899 minus 2 162119899 202119899 minus 9of linear and branched alkanes and in the strain energy ofcycloalkanes [25] Weighted entropy is a generalization ofShannonrsquos entropy and is themeasure of information suppliedby a probablistic experiment whose elementary events arecharacterized both by their objective probabilities and bysome qualitative (objective or subjective) weights [29] Itis useful to rank chemicals in quantitative high-throughputscreening experiments [30] and may be used to balancethe amount of information and the degree of homogeneityassociated to a partition of data in classes [31] Weightedentropy also found applications in the coding theory [32] Formore insights about applications of entropy please see [33]In this paper we have studied weighted entropy with atomicbond connectivity edge weights which was an open problemof [34] Our next aim is to work on entropy of weightedgraphs with geometric arithmetic and sum connectivity edgeweights
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
Authors do not have any competing interests
Authorsrsquo Contributions
All authors contributed equally to this paper
Acknowledgments
This work was supported by the Dong-A University researchfund
References
[1] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 2016
[2] W Gao M R Farahani and L Shi ldquoThe forgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterranea vol32 no 1 pp 579ndash585 2016
[3] S Kang Z Iqbal M Ishaq R Sarfraz A Aslam andW NazeerldquoOn eccentricity-based topological indices and polynomials ofphosphorus-containing dendrimersrdquo Symmetry vol 10 no 7 p237 2018
[4] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe degree-based topological indices of dendrimersrdquoChemistryvol 16 no 1 pp 681ndash688 2018
[5] S M Kang W Nazeer M A Zahid A R Nizami A Aslamand M Munir ldquoM-polynomials and topological indices of hex-derived networksrdquo Physics vol 16 no 1 pp 394ndash403 2018
[6] HWiener ldquoStructural determination of paraffinboiling pointsrdquoJournal of the American Chemical Society vol 69 no 1 pp 17ndash20 1947
[7] M Randic ldquoOn characterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23 pp6609ndash6615 1975
[8] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity indexmodeling the enthalpy of formation ofalkanesrdquo Indian Journal of Chemistry vol 37 pp 849ndash855 1998
[9] M Ghorbani and M A Hosseinzadeh ldquoHosseinzadeh Com-puting index of nanostar dendrimers Opto-electronrdquoOptoelec-tronics and Advanced Materials Rapid Communications vol 4no 9 pp 1419ndash1422 2010
[10] W Nazeer A Farooq M Younas M Munir and S Kang ldquoOnMolecular Descriptors of Carbon Nanoconesrdquo Biomoleculesvol 8 no 3 p 92 2018
[11] W Gao M Younas A Farooq A Mahboob and W NazeerldquoM-Polynomials and Degree-Based Topological Indices of theCrystallographic Structure of Moleculesrdquo Biomolecules vol 8no 4 p 107 2018
[12] Y C Kwun M Munir W Nazeer S Rafique and S M KangldquoComputational Analysis of topological indices of two BoronNanotubesrdquo Scientific reports vol 8 no 1 p 14843 2018
[13] W Gao M Younas A Farooq A Virk and W Nazeer ldquoSomeReverse Degree-Based Topological Indices and Polynomials ofDendrimersrdquoMathematics vol 6 no 10 p 214 2018
[14] M Munir W Nazeer S Rafique A R Nizami and S M KangldquoSome computational aspects of boron triangular nanotubesrdquoSymmetry vol 9 no 1 2017
[15] Y C Kwun M Munir W Nazeer S Rafique and S MKang ldquoM-Polynomials and topological indices of V-PhenylenicNanotubes and Nanotorirdquo Scientific Reports vol 7 no 1 2017
[16] C E Shannon The Mathematical Theory of CommunicationThe University of Illinois Press Urbana Ill USA 1949
[17] A Mowshowitz ldquoEntropy and the complexity of graphs II Theinformation content of digraphs and infinite graphsrdquo Bulletin ofMathematical Biology vol 30 pp 225ndash240 1968
[18] A Mowshowitz ldquoEntropy and the complexity of graphs IAn index of the relative complexity of a graphrdquo Bulletin ofMathematical Biology vol 30 pp 175ndash204 1968
[19] J Korner ldquoCoding of an information source having ambiguousalphabet and the entropy of graphsrdquo the 6th Pargue Conferenceon Information Theory Statistical Decision Functions RandomProcesses Pargue Czech Republic pp 411ndash425 1973
10 Discrete Dynamics in Nature and Society
[20] M M Dehmer N N Barbarini K K Varmuza and A AGraber ldquoNovel topological descriptors for analyzing biologicalnetworksrdquo BMC Structural Biology vol 10 article no 18 2010
[21] M Dehmer and A Mowshowitz ldquoA history of graph entropymeasuresrdquo Information Sciences vol 181 no 1 pp 57ndash78 2011
[22] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropybounds for dendrimersrdquo Applied Mathematics and Computa-tion vol 242 pp 462ndash472 2014
[23] S Ji X Li and B Huo ldquoOn reformulated Zagreb indices withrespect to acyclic unicyclic and bicyclic graphsrdquo The Matchvol 72 no 3 pp 723ndash732 2014
[24] K Xu K C Das and S Balachandran ldquoMaximizing theZagreb Indices of (nm)-Graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 72 pp 641ndash6542014
[25] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquo Filomat vol 26 no 4 pp 733ndash738 2012
[26] Y Bashir A AslamM Kamran et al ldquoOn forgotten topologicalindices of some dendrimers structurerdquoMolecules vol 22 no 62017
[27] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe Degree-based Topological Indices of Dendrimersrdquo OpenChemistry 2018
[28] A Aslam Y Bashir M Rafiq F Haider N MuhammadandN Bibi ldquoThree NewOld Vertex-Degree-Based TopologicalIndices of Some Dendrimers Structurerdquo Journal of Biology vol13 no 1 2017
[29] S Guiasu ldquoWeighted entropyrdquoReports onMathematical Physicsvol 2 no 3 pp 165ndash179 1971
[30] K R Shockley ldquoUsing weighted entropy to rank chemicals inquantitative high-throughput screening experimentsrdquo Journalof Biomolecular Screening vol 19 no 3 pp 344ndash353 2014
[31] S Guiasu ldquoGrouping data by using the weighted entropyrdquoJournal of Statistical Planning and Inference vol 15 no 1 pp63ndash69 1986
[32] A Clim ldquoWeighted entropy with applicationrdquo Analele Univer-sitatii Bucuresti Matematica pp 223ndash231 2008
[33] M Ghorbani M Dehmer and S Zangi ldquoGraph operationsbased on using distance-based graph entropiesrdquo Applied Math-ematics and Computation vol 333 pp 547ndash555 2018
[34] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropy ofweighted graphs with Randic weightsrdquo Entropy vol 17 no 6 pp3710ndash3723 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Discrete Dynamics in Nature and Society 9
Table 3 Edge partition of Zinc-Porphyrin dendrimers based on degree of end vertices of each edge
(119889119906119889V) (22) (23) (33) (3 4)Number of edges 162119899-4 402119899 minus 16 82119899 minus 16 4
Table 4 Edge partition of Poly(EThyleneAmidoAmine dendrimers based on degree of end vertices of each edge
(119889119906119889V) (12) (13) (22) (2 3)Number of edges 42119899 42119899 minus 2 162119899 202119899 minus 9of linear and branched alkanes and in the strain energy ofcycloalkanes [25] Weighted entropy is a generalization ofShannonrsquos entropy and is themeasure of information suppliedby a probablistic experiment whose elementary events arecharacterized both by their objective probabilities and bysome qualitative (objective or subjective) weights [29] Itis useful to rank chemicals in quantitative high-throughputscreening experiments [30] and may be used to balancethe amount of information and the degree of homogeneityassociated to a partition of data in classes [31] Weightedentropy also found applications in the coding theory [32] Formore insights about applications of entropy please see [33]In this paper we have studied weighted entropy with atomicbond connectivity edge weights which was an open problemof [34] Our next aim is to work on entropy of weightedgraphs with geometric arithmetic and sum connectivity edgeweights
Data Availability
The data used to support the findings of this study areincluded within the article
Conflicts of Interest
Authors do not have any competing interests
Authorsrsquo Contributions
All authors contributed equally to this paper
Acknowledgments
This work was supported by the Dong-A University researchfund
References
[1] W Gao W Wang and M R Farahani ldquoTopological indicesstudy of molecular structure in anticancer drugsrdquo Journal ofChemistry vol 2016 2016
[2] W Gao M R Farahani and L Shi ldquoThe forgotten topologicalindex of some drug structuresrdquo Acta Medica Mediterranea vol32 no 1 pp 579ndash585 2016
[3] S Kang Z Iqbal M Ishaq R Sarfraz A Aslam andW NazeerldquoOn eccentricity-based topological indices and polynomials ofphosphorus-containing dendrimersrdquo Symmetry vol 10 no 7 p237 2018
[4] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe degree-based topological indices of dendrimersrdquoChemistryvol 16 no 1 pp 681ndash688 2018
[5] S M Kang W Nazeer M A Zahid A R Nizami A Aslamand M Munir ldquoM-polynomials and topological indices of hex-derived networksrdquo Physics vol 16 no 1 pp 394ndash403 2018
[6] HWiener ldquoStructural determination of paraffinboiling pointsrdquoJournal of the American Chemical Society vol 69 no 1 pp 17ndash20 1947
[7] M Randic ldquoOn characterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23 pp6609ndash6615 1975
[8] E Estrada L Torres L Rodrıguez and I Gutman ldquoAn atom-bond connectivity indexmodeling the enthalpy of formation ofalkanesrdquo Indian Journal of Chemistry vol 37 pp 849ndash855 1998
[9] M Ghorbani and M A Hosseinzadeh ldquoHosseinzadeh Com-puting index of nanostar dendrimers Opto-electronrdquoOptoelec-tronics and Advanced Materials Rapid Communications vol 4no 9 pp 1419ndash1422 2010
[10] W Nazeer A Farooq M Younas M Munir and S Kang ldquoOnMolecular Descriptors of Carbon Nanoconesrdquo Biomoleculesvol 8 no 3 p 92 2018
[11] W Gao M Younas A Farooq A Mahboob and W NazeerldquoM-Polynomials and Degree-Based Topological Indices of theCrystallographic Structure of Moleculesrdquo Biomolecules vol 8no 4 p 107 2018
[12] Y C Kwun M Munir W Nazeer S Rafique and S M KangldquoComputational Analysis of topological indices of two BoronNanotubesrdquo Scientific reports vol 8 no 1 p 14843 2018
[13] W Gao M Younas A Farooq A Virk and W Nazeer ldquoSomeReverse Degree-Based Topological Indices and Polynomials ofDendrimersrdquoMathematics vol 6 no 10 p 214 2018
[14] M Munir W Nazeer S Rafique A R Nizami and S M KangldquoSome computational aspects of boron triangular nanotubesrdquoSymmetry vol 9 no 1 2017
[15] Y C Kwun M Munir W Nazeer S Rafique and S MKang ldquoM-Polynomials and topological indices of V-PhenylenicNanotubes and Nanotorirdquo Scientific Reports vol 7 no 1 2017
[16] C E Shannon The Mathematical Theory of CommunicationThe University of Illinois Press Urbana Ill USA 1949
[17] A Mowshowitz ldquoEntropy and the complexity of graphs II Theinformation content of digraphs and infinite graphsrdquo Bulletin ofMathematical Biology vol 30 pp 225ndash240 1968
[18] A Mowshowitz ldquoEntropy and the complexity of graphs IAn index of the relative complexity of a graphrdquo Bulletin ofMathematical Biology vol 30 pp 175ndash204 1968
[19] J Korner ldquoCoding of an information source having ambiguousalphabet and the entropy of graphsrdquo the 6th Pargue Conferenceon Information Theory Statistical Decision Functions RandomProcesses Pargue Czech Republic pp 411ndash425 1973
10 Discrete Dynamics in Nature and Society
[20] M M Dehmer N N Barbarini K K Varmuza and A AGraber ldquoNovel topological descriptors for analyzing biologicalnetworksrdquo BMC Structural Biology vol 10 article no 18 2010
[21] M Dehmer and A Mowshowitz ldquoA history of graph entropymeasuresrdquo Information Sciences vol 181 no 1 pp 57ndash78 2011
[22] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropybounds for dendrimersrdquo Applied Mathematics and Computa-tion vol 242 pp 462ndash472 2014
[23] S Ji X Li and B Huo ldquoOn reformulated Zagreb indices withrespect to acyclic unicyclic and bicyclic graphsrdquo The Matchvol 72 no 3 pp 723ndash732 2014
[24] K Xu K C Das and S Balachandran ldquoMaximizing theZagreb Indices of (nm)-Graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 72 pp 641ndash6542014
[25] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquo Filomat vol 26 no 4 pp 733ndash738 2012
[26] Y Bashir A AslamM Kamran et al ldquoOn forgotten topologicalindices of some dendrimers structurerdquoMolecules vol 22 no 62017
[27] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe Degree-based Topological Indices of Dendrimersrdquo OpenChemistry 2018
[28] A Aslam Y Bashir M Rafiq F Haider N MuhammadandN Bibi ldquoThree NewOld Vertex-Degree-Based TopologicalIndices of Some Dendrimers Structurerdquo Journal of Biology vol13 no 1 2017
[29] S Guiasu ldquoWeighted entropyrdquoReports onMathematical Physicsvol 2 no 3 pp 165ndash179 1971
[30] K R Shockley ldquoUsing weighted entropy to rank chemicals inquantitative high-throughput screening experimentsrdquo Journalof Biomolecular Screening vol 19 no 3 pp 344ndash353 2014
[31] S Guiasu ldquoGrouping data by using the weighted entropyrdquoJournal of Statistical Planning and Inference vol 15 no 1 pp63ndash69 1986
[32] A Clim ldquoWeighted entropy with applicationrdquo Analele Univer-sitatii Bucuresti Matematica pp 223ndash231 2008
[33] M Ghorbani M Dehmer and S Zangi ldquoGraph operationsbased on using distance-based graph entropiesrdquo Applied Math-ematics and Computation vol 333 pp 547ndash555 2018
[34] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropy ofweighted graphs with Randic weightsrdquo Entropy vol 17 no 6 pp3710ndash3723 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
10 Discrete Dynamics in Nature and Society
[20] M M Dehmer N N Barbarini K K Varmuza and A AGraber ldquoNovel topological descriptors for analyzing biologicalnetworksrdquo BMC Structural Biology vol 10 article no 18 2010
[21] M Dehmer and A Mowshowitz ldquoA history of graph entropymeasuresrdquo Information Sciences vol 181 no 1 pp 57ndash78 2011
[22] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropybounds for dendrimersrdquo Applied Mathematics and Computa-tion vol 242 pp 462ndash472 2014
[23] S Ji X Li and B Huo ldquoOn reformulated Zagreb indices withrespect to acyclic unicyclic and bicyclic graphsrdquo The Matchvol 72 no 3 pp 723ndash732 2014
[24] K Xu K C Das and S Balachandran ldquoMaximizing theZagreb Indices of (nm)-Graphsrdquo MATCH Communications inMathematical and in Computer Chemistry vol 72 pp 641ndash6542014
[25] K C Das I Gutman and B Furtula ldquoOn atom-bond connec-tivity indexrdquo Filomat vol 26 no 4 pp 733ndash738 2012
[26] Y Bashir A AslamM Kamran et al ldquoOn forgotten topologicalindices of some dendrimers structurerdquoMolecules vol 22 no 62017
[27] S M Kang M A Zahid W Nazeer and W Gao ldquoCalculatingthe Degree-based Topological Indices of Dendrimersrdquo OpenChemistry 2018
[28] A Aslam Y Bashir M Rafiq F Haider N MuhammadandN Bibi ldquoThree NewOld Vertex-Degree-Based TopologicalIndices of Some Dendrimers Structurerdquo Journal of Biology vol13 no 1 2017
[29] S Guiasu ldquoWeighted entropyrdquoReports onMathematical Physicsvol 2 no 3 pp 165ndash179 1971
[30] K R Shockley ldquoUsing weighted entropy to rank chemicals inquantitative high-throughput screening experimentsrdquo Journalof Biomolecular Screening vol 19 no 3 pp 344ndash353 2014
[31] S Guiasu ldquoGrouping data by using the weighted entropyrdquoJournal of Statistical Planning and Inference vol 15 no 1 pp63ndash69 1986
[32] A Clim ldquoWeighted entropy with applicationrdquo Analele Univer-sitatii Bucuresti Matematica pp 223ndash231 2008
[33] M Ghorbani M Dehmer and S Zangi ldquoGraph operationsbased on using distance-based graph entropiesrdquo Applied Math-ematics and Computation vol 333 pp 547ndash555 2018
[34] Z Chen M Dehmer F Emmert-Streib and Y Shi ldquoEntropy ofweighted graphs with Randic weightsrdquo Entropy vol 17 no 6 pp3710ndash3723 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
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